A208513 Triangle of coefficients of polynomials u(n,x) jointly generated with A111125; see the Formula section.

1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 20, 8, 1, 1, 25, 50, 35, 10, 1, 1, 36, 105, 112, 54, 12, 1, 1, 49, 196, 294, 210, 77, 14, 1, 1, 64, 336, 672, 660, 352, 104, 16, 1, 1, 81, 540, 1386, 1782, 1287, 546, 135, 18, 1, 1, 100, 825, 2640, 4290, 4004, 2275, 800, 170, 20, 1

Offset: 1

Author

Clark Kimberling, Feb 28 2012

Keywords

• nonn
• tabl

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)

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Crossrefs

Cf. A085478, A078812, A111125, A156308, A208509, A211956.

Cf. A127677, A263916.

Programs

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

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 *)

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

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