A109449 - OEIS

A109449

Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 8, 6, 4, 1, 16, 25, 20, 10, 5, 1, 61, 96, 75, 40, 15, 6, 1, 272, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1, 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1, 50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1

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OFFSET

0,5

COMMENTS

The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k=0..n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.

The 'signed version' of the triangle is the exponential Riordan array [sech(x)+tanh(x), x]. - Peter Luschny, Jan 24 2009

Up to signs, the matrix is self-inverse: T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 15 2013

LINKS

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Peter Luschny, The Swiss-Knife polynomials.

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

Wikipedia, Boustrophedon transform

Index entries for sequences related to boustrophedon transform

FORMULA

Sum_{k>=0} T(n, k) = A000667(n).

Sum_{k>=0} T(2n, 2k) = A000795(n).

Sum_{k>=0} T(2n, 2k+1) = A009747(n).

Sum_{k>=0} T(2n+1, 2k) = A003719(n).

Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).

Sum_{k>=0} T(n, 2k) = A062272(n).

Sum_{k>=0} T(n, 2k+1) = A062161(n).

Sum_{k>=0} (-1)^(k)*T(n, k) = A062162(n). - Johannes W. Meijer, Apr 20 2011

E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic, May 20 2007

T(n,k) = 2^(n-k)C(n,k)|E(n-k,1/2)+E(n-k,1)|-[n=k] where C(n,k) is the binomial coefficient, E(m,x) are the Euler polynomials and [] the Iverson bracket. - Peter Luschny, Jan 24 2009

From Reikku Kulon, Feb 26 2009: (Start)

A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).

A109449(n, n - 1) = n.

A109449(n, n) = 1.

For n > 0, k > 0: A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)

From Peter Luschny, Jul 10 2009: (Start)

Let p_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v C(k,v)F(k)(x+v+1)^n, where F(0)=1 and for k>0 F(k)=-1 + s_k 2^floor((k-1)/2), s_k is 0 if k mod 8 in {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute values of the coefficients of these polynomials.

Another way to express the polynomials p_n(x) is

p_n(x) = -x^n + Sum_{k=0..n} binomial(n,k) Euler(k)((x+1)^(n-k)+x^(n-k)). (End)

From Peter Bala, Jan 26 2011: (Start)

An explicit formula for the n-th row polynomial is

x^n + i*Sum_{k=1..n}((1+i)/2)^(k-1)*Sum_{j=0..k} (-1)^j*binomial(k,j)*(x+i*j)^n, where i = sqrt(-1). This is the triangle of connection constants between the polynomial sequences {Z(n,x+1)} and {Z(n,x)}, where Z(n,x) denotes the zigzag polynomials described in A147309.

Denote the present array by M. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of A145876 - the generalized Eulerian numbers associated with the zigzag numbers. (End)

Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then

T(n,k) = [x^(n-k)] |skp{n}(x) - skp{n}(x-1) + x^n|. - Peter Luschny, Jul 22 2012

T(n,k) = A007318(n,k) * A000111(n - k), k = 0..n. - Reinhard Zumkeller, Nov 02 2013

T(n,k) = abs(A247453(n,k)). - Reinhard Zumkeller, Sep 17 2014

EXAMPLE

Triangle starts:

1, 1;

1, 2, 1;

2, 3, 3, 1;

5, 8, 6, 4, 1;

16, 25, 20, 10, 5, 1;

61, 96, 75, 40, 15, 6, 1;

272, 427, 336, 175, 70, 21, 7, 1;

1385, 2176, 1708, 896, 350, 112, 28, 8, 1;

7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1;

50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...

MAPLE

From Peter Luschny, Jul 10 2009, edited Jun 06 2022: (Start)

A109449 := (n, k) -> binomial(n, k)*A000111(n-k):

seq(print(seq(A109449(n, k), k=0..n)), n=0..9);

B109449 := (n, k) -> 2^(n-k)*binomial(n, k)*abs(euler(n-k, 1/2)+euler(n-k, 1)) -`if`(n-k=0, 1, 0): seq(print(seq(B109449(n, k), k=0..n)), n=0..9);

R109449 := proc(n, k) option remember; if k = 0 then A000111(n) else R109449(n-1, k-1)*n/k fi end: seq(print(seq(R109449(n, k), k=0..n)), n=0..9);

E109449 := proc(n) add(binomial(n, k)*euler(k)*((x+1)^(n-k)+ x^(n-k)), k=0..n) -x^n end: seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);

sigma := n -> ifelse(n=0, 1, [1, 1, 0, -1, -1, -1, 0, 1][n mod 8 + 1]/2^iquo(n-1, 2)-1):

L109449 := proc(n) add(add((-1)^v*binomial(k, v)*(x+v+1)^n*sigma(k), v=0..k), k=0..n) end: seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);

X109449 := n -> n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n): seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9);

(End)

MATHEMATICA

lim = 10; s = CoefficientList[Series[(1 + Sin[x])/Cos[x], {x, 0, lim}], x] Table[k!, {k, 0, lim}]; Table[Binomial[n, k] s[[n - k + 1]], {n, 0, lim}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015, after Jean-François Alcover at A000111 *)

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 27 2019 *)

PROG

(Sage)

R = PolynomialRing(ZZ, 'x')

@CachedFunction

def skp(n, x) :

if n == 0 : return 1

return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])

def A109449_row(n):

x = R.gen()

return [abs(c) for c in list(skp(n, x)-skp(n, x-1)+x^n)]

for n in (0..10) : print(A109449_row(n)) # Peter Luschny, Jul 22 2012

(Haskell)