A321507 - OEIS

%I #7 Nov 14 2018 14:01:38

%S 1,1,1,3,3,3,10,10,10,22,29,29,56,70,70,127,176,176,283,367,395,644,

%T 833,889,1315,1714,1910,2791,3606,3942,5538,7413,8169,11100,14544,

%U 16140,21927,28886,32344,42152,54728,62624,81625,105148,120310,152699,197624

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

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e 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}].

%t 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}]

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

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Nov 11 2018