A363778 -id:A363778 - cp's OEIS frontend

A363779 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^3))^k.

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 0, 0, 1, -9, 36, -84, 126, -126, 84, -36, 7, 1, 0, 1, -10, 45, -120, 210, -252, 210, -120, 42, -4, -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,  1,  5,  15,  35,   70,  126, ...
  0, -1, -6, -21, -56, -126, -252, ...
  0,  1,  7,  28,  84,  210,  462, ...

Columns k=0..3 give A000007, A323633, A363776, A363777.
Main diagonal gives A363781.
Cf. A290054, A363778, A363783.

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A363783(j) * T(n-j,k).

A363774 Expansion of 1/(Sum_{k>=0} x^(k^2))^2.

Original entry on oeis.org

1, -2, 3, -4, 3, 0, -5, 12, -18, 18, -9, -12, 44, -76, 93, -76, 5, 120, -273, 400, -414, 228, 200, -828, 1480, -1842, 1539, -268, -2004, 4824, -7168, 7568, -4518, -2784, 13577, -24900, 31563, -27236, 6816, 30308, -77010, 116844, -126018, 80180, 34140, -205932, 389275

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A000925.
Column k=2 of A363778.
Cf. A162552, A274621.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, sqrtint(N), x^k^2)^2)

a(0) = 1; a(n) = -(2/n) * Sum_{k=1..n} A162552(k) * a(n-k).

A363775 Expansion of 1/(Sum_{k>=0} x^(k^2))^3.

Original entry on oeis.org

1, -3, 6, -10, 12, -9, -2, 24, -54, 80, -84, 42, 66, -234, 420, -536, 450, -39, -740, 1770, -2688, 2898, -1722, -1320, 6078, -11349, 14736, -12992, 3084, 15999, -41212, 64032, -70788, 46020, 20778, -126132, 244120, -323421, 295410, -96848, -293868, 815829, -1297972

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A002102.
Column k=3 of A363778.
Cf. A162552.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, sqrtint(N), x^k^2)^3)

a(0) = 1; a(n) = -(3/n) * Sum_{k=1..n} A162552(k) * a(n-k).

A363780 a(n) = [x^n] 1/(Sum_{k>=0} x^(k^2))^n.

Original entry on oeis.org

1, -1, 3, -10, 31, -96, 294, -876, 2511, -6796, 16698, -33540, 31174, 184534, -1627812, 8912760, -41466433, 176963760, -714194382, 2766892840, -10374065814, 37815483948, -134334781732, 465432203640, -1571910265770, 5164302815179, -16438631981418

Offset: 0

Seiichi Manyama, Jun 21 2023

sign

Main diagonal of A363778.

Cf. A287617.

cp's OEIS Frontend

A363779 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^3))^k.

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A363774 Expansion of 1/(Sum_{k>=0} x^(k^2))^2.

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Formula

A363775 Expansion of 1/(Sum_{k>=0} x^(k^2))^3.

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Programs

PARI

Formula

A363780 a(n) = [x^n] 1/(Sum_{k>=0} x^(k^2))^n.

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