A321507 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^A072964(k).

1, 1, 1, 3, 3, 3, 10, 10, 10, 22, 29, 29, 56, 70, 70, 127, 176, 176, 283, 367, 395, 644, 833, 889, 1315, 1714, 1910, 2791, 3606, 3942, 5538, 7413, 8169, 11100, 14544, 16140, 21927, 28886, 32344, 42152, 54728, 62624, 81625, 105148, 120310, 152699, 197624

Offset

0,4

Comments

a(n) is the number of partitions of n into triangular numbers k*(k + 1)/2 of A072964(k) kinds.

Links

Table of n, a(n) for n=0..46.

Index entries for sequences related to partitions

Formula

G.f.: Product_{k>=1} 1/(1 - x^A000217(k))^A007294(A000217(k)).

Example

a(6) = 10 because we have [{6}], [{3, 3}],  [{3}, {3}], [{3, 1, 1, 1}], [{3}, {1, 1, 1}], [{3}, {1}, {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}].

Mathematica

b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - x^(k (k + 1)/2)), {k, 1, n}], {x, 0, n (n + 1)/2}]; a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^(k (k + 1)/2))^b[k], {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 0, 46}]

Crossrefs

Cf. A000217, A001970, A007294, A072964, A300300.

Sequence in context: A097707 A240533 A115282 * A147605 A029616 A278822

Adjacent sequences: A321504 A321505 A321506 * A321508 A321509 A321510

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 11 2018

Status

approved