A320846 Expansion of Product_{k>=1} 1/(1 - x^(k^2))^A037444(k).

1, 1, 1, 1, 3, 3, 3, 3, 6, 10, 10, 10, 14, 22, 22, 22, 35, 47, 57, 57, 79, 95, 115, 115, 146, 217, 247, 267, 307, 433, 473, 513, 598, 779, 985, 1045, 1253, 1489, 1861, 1941, 2272, 2859, 3397, 3847, 4301, 5467, 6171, 6991, 7688, 9531, 11559, 12749, 14693

Offset

0,5

Comments

a(n) is the number of partitions of n into squares k^2 of A037444(k) kinds.

Links

Table of n, a(n) for n=0..52.

Index entries for sequences related to partitions

Formula

G.f.: Product_{k>=1} 1/(1 - x^A000290(k))^A001156(A000290(k)).

Example

a(8) = 6 because we have [{4}, {4}], [{4}, {1, 1, 1, 1}], [{4}, {1}, {1}, {1}, {1}], [{1, 1, 1, 1}, {1, 1, 1, 1}], [{1, 1, 1, 1}, {1}, {1}, {1}, {1}] and [{1}, {1}, {1}, {1}, {1}, {1}, {1}, {1}].

Mathematica

b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - x^k^2), {k, 1, n}], {x, 0, n^2}]; a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^k^2)^b[k], {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 0, 52}]

Crossrefs

Cf. A000290, A001156, A001970, A037444, A045842, A285047, A300300.

Sequence in context: A289903 A285047 A263137 * A277515 A247202 A195758

Adjacent sequences: A320843 A320844 A320845 * A320847 A320848 A320849

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 11 2018

Status

approved