A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+...+(n+1)*x^n)^2.

1, 1, 4, 4, 1, 4, 10, 12, 9, 1, 4, 10, 20, 25, 24, 16, 1, 4, 10, 20, 35, 44, 46, 40, 25, 1, 4, 10, 20, 35, 56, 70, 76, 73, 60, 36, 1, 4, 10, 20, 35, 56, 84, 104, 115, 116, 106, 84, 49, 1, 4, 10, 20, 35, 56, 84, 120, 147, 164, 170, 164, 145, 112, 64

Offset: 0

Author

Seiichi Manyama, Feb 29 2020

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1, 4,  4;
  1, 4, 10, 12,  9;
  1, 4, 10, 20, 25, 24, 16;
  1, 4, 10, 20, 35, 44, 46, 40, 25;
  ...
		

Links

• Seiichi Manyama, Rows n = 0..99, flattened

Crossrefs

Row sums give A000537(n+1).

T(n,2n) gives A000290(n+1).

Cf. A000292, A002260, A084608, A008794, A329709.

Programs

Mathematica

row[n_]:=CoefficientList[Series[(Sum[(i+1)x^i,{i,0,n}])^2,{x,0,2n}],x]; Array[row,8,0]//Flatten (* Stefano Spezia, Feb 15 2025 *)

PARI

for(n=0, 10, print(Vecrev(sum(k=0, n, (k+1)*x^k)^2), ", "))

Formula

T(n,k) = A000292(k+1) for k=0..n.

Sum_{k=0..2n} (-1)^k * T(n,k) = A008794(n+2). - Alois P. Heinz, Feb 14 2025