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A288126 Number of partitions of n-th triangular number (A000217) into distinct triangular parts.

%I A288126 #12 Feb 16 2025 08:33:47
%S A288126 1,1,1,1,2,1,2,3,2,4,7,6,4,14,15,19,31,28,43,57,80,103,127,181,234,
%T A288126 295,398,539,663,888,1178,1419,1959,2519,3102,4201,5282,6510,8717,
%U A288126 11162,13557,18108,22965,28206,36860,46350,58060,73857,93541,117058,147376,186158,232949,292798,365639
%N A288126 Number of partitions of n-th triangular number (A000217) into distinct triangular parts.
%H A288126 Robert Israel, <a href="/A288126/b288126.txt">Table of n, a(n) for n = 0..1000</a>
%H A288126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A288126 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H A288126 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A288126 a(n) = [x^(n*(n+1)/2)] Product_{k>=1} (1 + x^(k(k+1)/2)).
%F A288126 a(n) = A024940(A000217(n)).
%e A288126 a(4) = 2 because 4th triangular number is 10 and we have [10], [6, 3, 1].
%p A288126 N:= 100:
%p A288126 G:= mul(1+x^(k*(k+1)/2),k=1..N):
%p A288126 seq(coeff(G,x,n*(n+1)/2),n=0..N); # _Robert Israel_, Jun 06 2017
%t A288126 Table[SeriesCoefficient[Product[1 + x^(k (k + 1)/2), {k, 1, n}], {x, 0, n (n + 1)/2}], {n, 0, 54}]
%Y A288126 Cf. A000217, A007294, A024940, A030273, A037444, A072964.
%K A288126 nonn
%O A288126 0,5
%A A288126 _Ilya Gutkovskiy_, Jun 05 2017