A329708 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

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

EXAMPLE

Triangle begins:

1, 4, 4;

1, 4, 10, 12, 9;

1, 4, 10, 20, 25, 24, 16;

1, 4, 10, 20, 35, 44, 46, 40, 25;

...

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 *)

PROG

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

CROSSREFS

Row sums give A000537(n+1).

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

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

Sequence in context: A213056 A135012 A156380 * A263493 A166237 A021878

Adjacent sequences: A329705 A329706 A329707 * A329709 A329710 A329711

KEYWORD

nonn,tabf

AUTHOR

Seiichi Manyama, Feb 29 2020

STATUS

approved