A288126 - OEIS

%I #12 Feb 16 2025 08:33:47

%S 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 295,398,539,663,888,1178,1419,1959,2519,3102,4201,5282,6510,8717,

%U 11162,13557,18108,22965,28206,36860,46350,58060,73857,93541,117058,147376,186158,232949,292798,365639

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

%H Robert Israel, <a href="/A288126/b288126.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>

%F a(n) = [x^(n*(n+1)/2)] Product_{k>=1} (1 + x^(k(k+1)/2)).

%F a(n) = A024940(A000217(n)).

%e a(4) = 2 because 4th triangular number is 10 and we have [10], [6, 3, 1].

%p N:= 100:

%p G:= mul(1+x^(k*(k+1)/2),k=1..N):

%p seq(coeff(G,x,n*(n+1)/2),n=0..N); # _Robert Israel_, Jun 06 2017

%t Table[SeriesCoefficient[Product[1 + x^(k (k + 1)/2), {k, 1, n}], {x, 0, n (n + 1)/2}], {n, 0, 54}]

%Y Cf. A000217, A007294, A024940, A030273, A037444, A072964.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Jun 05 2017