A363783 -id:A363783 - 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).

A363784 L.g.f.: log( Sum_{k>=0} x^(k^4) ).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 15, 1, -18, 36, -55, 75, -96, 118, -141, 165, -190, 216, -243, 271, -300, 330, -345, 344, -326, 290, -235, 160, -64, -54, 195, -360, 550, -766, 1009, -1280, 1580

Offset: 1

Seiichi Manyama, Jun 21 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A162552, A363783.

L.g.f.: L(x) = Sum_{k>=1} a(k)*x^k/k = log( Sum_{k>=0} x^(k^4) ).

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).

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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A363784 L.g.f.: log( Sum_{k>=0} x^(k^4) ).

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

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

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