A304631 -id:A304631 - cp's OEIS frontend

A304632 Expansion of (1/(1 - x))* Product_{k>=1} (1 + x^k)/(1 + x^(3*k)).

1, 2, 3, 4, 5, 7, 9, 12, 15, 18, 22, 27, 33, 40, 48, 57, 67, 79, 93, 109, 127, 147, 170, 196, 226, 260, 298, 340, 387, 440, 500, 567, 641, 723, 814, 916, 1030, 1156, 1295, 1448, 1617, 1804, 2011, 2239, 2489, 2763, 3064, 3395, 3759, 4158, 4594, 5070, 5590, 6159, 6781, 7460, 8199, 9003

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Partial sums of A003105.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Schur's Partition Theorem
Index entries for sequences related to partitions

Cf. A000070, A003105, A036469, A304630, A304631.

nmax = 57; CoefficientList[Series[1/(1 - x) Product[(1 + x^k)/(1 + x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 57; CoefficientList[Series[1/(1 - x) Product[1/((1 - x^(6 k + 1)) (1 - x^(6 k + 5))), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 57; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k + x^(2 k)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=0} 1/((1 - x^(6*k+1))*(1 - x^(6*k+5))).
G.f.: (1/(1 - x))*Product_{k>=0} 1/(1 - x^k + x^(2*k)).
a(n) ~ exp(sqrt(2*n)*Pi/3) * sqrt(3) / (Pi * 2^(3/4) * n^(1/4)). - Vaclav Kotesovec, May 19 2018

A382974 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{j>=1} 1/(1 - x^j + y^j).

Original entry on oeis.org

1, -1, 1, 0, -2, 2, -1, 2, -4, 3, 1, -3, 4, -7, 5, -1, 4, -8, 10, -12, 7, 1, -5, 14, -20, 18, -19, 11, -1, 6, -18, 34, -40, 34, -30, 15, 2, -7, 22, -51, 78, -77, 56, -45, 22, -2, 9, -30, 75, -127, 157, -139, 94, -67, 30, 2, -11, 42, -105, 196, -282, 306, -239, 146, -97, 42

Offset: 0

Seiichi Manyama, Apr 11 2025

			Square array begins:
   1,  -1,  0,   -1,   1,   -1,    1, ...
   1,  -2,  2,   -3,   4,   -5,    6, ...
   2,  -4,  4,   -8,  14,  -18,   22, ...
   3,  -7, 10,  -20,  34,  -51,   75, ...
   5, -12, 18,  -40,  78, -127,  196, ...
   7, -19, 34,  -77, 157, -282,  478, ...
  11, -30, 56, -139, 306, -582, 1048, ...

Columns k=0..1 give A000041, (-1)*A000070.
Rows n=0..1 give A081362, (-1)^k * A304631(k).
Main diagonal gives A382979.
Antidiagonal sums give A000007.
Cf. A322210.

cp's OEIS Frontend

A304632 Expansion of (1/(1 - x))* Product_{k>=1} (1 + x^k)/(1 + x^(3*k)).

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A382974 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = [x^n * y^k] Product_{j>=1} 1/(1 - x^j + y^j).

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