OFFSET
1,5
COMMENTS
The columns of A208513 are identical to those of A208509. Here, however, the alternating row sums are periodic (with period 1,0,-2,-3,-2,0).
From Tom Copeland, Nov 07 2015: (Start)
These polynomials may be expressed in terms of the Faber polynomials of A263916, similar to A127677.
Rephrasing notes in A111125: Append an initial column of zeros except for a 1 at the top to A111125. Then the rows of this entry contain the partial sums of the column sequences of modified A111125; therefore, the difference of consecutive pairs of rows of this entry, modified by appending an initial row of zeros to it, generates the modified A111125. (End)
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials
FORMULA
Coefficients of u(n, x) from the mixed recurrence relations:
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x) = 1, u(2,x) = 1+x, v(1,x) = 1, v(2,x) = 3+x.
From Peter Bala, May 01 2012: (Start)
Working with an offset of 0: T(n,0) = 1; T(n,k) = (n/k)*binomial(n+k-1,2*k-1) = (n/k)*A078812(n,k) for k > 0. Cf. A156308.
O.g.f.: ((1-t)^2 + t^2*x)/((1-t)*((1-t)^2-t*x)) = 1 + (1+x)*t + (1+4*x+x^2)*t^2 + ....
u(n+1,x) = -1 + (b(2*n,x) + 1)/b(n,x), where b(n,x) = Sum_{k = 0..n} binomial(n+k, 2*k)*x^k are the Morgan-Voyce polynomials of A085478.
This triangle is formed from the even numbered rows of A211956 with a factor of 2^(k-1) removed from the k-th column entries.
(End)
T(n, k) = (2*(n-1)/(n+k-2))*binomial(n+k-2, 2*k-2). - G. C. Greubel, Feb 02 2022
EXAMPLE
First five rows:
1;
1, 1;
1, 4, 1;
1, 9, 6, 1;
1, 16, 20, 8, 1;
First five polynomials u(n,x):
u(1,x) = 1;
u(2,x) = 1 + x;
u(3,x) = 1 + 4*x + x^2;
u(4,x) = 1 + 9*x + 6*x^2 + x^3;
u(5,x) = 1 + 16*x + 20*x^2 + 8*x^3 + x^4;
MATHEMATICA
(* First program *)
u[1, x_]:=1; v[1, x_]:=1; z=16;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, z}];
TableForm[cu]
Flatten[%] (* A208513 *)
Table[Expand[v[n, x]], {n, z}]
cv = Table[CoefficientList[v[n, x], x], {n, z}];
TableForm[cv]
Flatten[%] (* A111125 *)
(* Second program *)
T[n_, k_]:= If[k==1, 1, ((n-1)/(k-1))*Binomial[n+k-3, 2*k-3]];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Feb 02 2022 *)
PROG
(Magma)
A208513:= func< n, k | k eq 1 select 1 else (2*(n-1)/(n+k-2))*Binomial(n+k-2, 2*k-2) >;
[A208513(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 02 2022
(Sage)
def A208513(n, k): return 1 if (k==1) else ((n-1)/(k-1))*binomial(n+k-3, 2*k-3)
flatten([[A208513(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 02 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 28 2012
STATUS
approved