A363779 -id:A363779 - cp's OEIS frontend

A363778 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^2))^k.

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 0, 0, 1, -5, 10, -10, 3, 1, 0, 1, -6, 15, -20, 12, 0, -2, 0, 1, -7, 21, -35, 31, -9, -5, 3, 0, 1, -8, 28, -56, 65, -36, -2, 12, -3, 0, 1, -9, 36, -84, 120, -96, 24, 24, -18, 1, 0, 1, -10, 45, -120, 203, -210, 105, 20, -54, 18, 2, 0

Offset: 0

Seiichi Manyama, Jun 21 2023

			Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0, -1, -2,  -3,  -4,  -5,   -6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0, -1, -4, -10, -20, -35,  -56, ...
  0,  0,  3,  12,  31,  65,  120, ...
  0,  1,  0,  -9, -36, -96, -210, ...
  0, -2, -5,  -2,  24, 105,  294, ...

Columns k=0..3 give A000007, A317665, A363774, A363775.
Main diagonal gives A363780.
Cf. A045847, A162552, A363779.

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A162552(j) * T(n-j,k).

A363776 Expansion of 1/(Sum_{k>=0} x^(k^3))^2.

Original entry on oeis.org

1, -2, 3, -4, 5, -6, 7, -8, 7, -4, -1, 8, -17, 28, -41, 56, -70, 80, -83, 76, -56, 20, 35, -112, 210, -324, 445, -562, 658, -712, 699, -590, 357, 22, -558, 1252, -2084, 3008, -3947, 4788, -5383, 5556, -5116, 3864, -1618, -1756, 6307, -11956, 18454, -25348, 31962, -37380

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A173677.
Column k=2 of A363779.
Cf. A363783.

my(N=60, x='x+O('x^N)); Vec(1/sum(k=0, N^(1/3), x^k^3)^2)

a(0) = 1; a(n) = -(2/n) * Sum_{k=1..n} A363783(k) * a(n-k).

A363777 Expansion of 1/(Sum_{k>=0} x^(k^3))^3.

Original entry on oeis.org

1, -3, 6, -10, 15, -21, 28, -36, 42, -43, 36, -18, -14, 63, -132, 224, -336, 459, -578, 672, -714, 671, -504, 168, 378, -1161, 2184, -3419, 4794, -6183, 7396, -8169, 8169, -7014, 4308, 321, -7156, 16311, -27636, 40607, -54222, 66924, -76572, 80451, -75372, 57883

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A051343.
Column k=3 of A363779.
Cf. A363783.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, N^(1/3), x^k^3)^3)

a(0) = 1; a(n) = -(3/n) * Sum_{k=1..n} A363783(k) * a(n-k).

A363781 a(n) = [x^n] 1/(Sum_{k>=0} x^(k^3))^n.

Original entry on oeis.org

1, -1, 3, -10, 35, -126, 462, -1716, 6427, -24220, 91718, -348712, 1330238, -5088916, 19515660, -75000600, 288772795, -1113688667, 4301387580, -16635049690, 64410064110, -249659926890, 968651123520, -3761601535620, 14619516646150, -56861702382576

Offset: 0

Seiichi Manyama, Jun 21 2023

sign

Main diagonal of A363779.

Cf. A291700.

cp's OEIS Frontend

A363778 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^2))^k.

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A363776 Expansion of 1/(Sum_{k>=0} x^(k^3))^2.

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Formula

A363777 Expansion of 1/(Sum_{k>=0} x^(k^3))^3.

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Programs

PARI

Formula

A363781 a(n) = [x^n] 1/(Sum_{k>=0} x^(k^3))^n.

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