A281083 - OEIS

%I #15 Feb 16 2025 08:33:39

%S 1,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,0,

%T 0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,

%U 1,0,0,0,0,1,1,0,1,1,0,0,0,0,2,2,0,0,0,0,1,1,0,0,1,1,0,0,0,0,2,2,0,0,0,0,1,1

%N Expansion of Product_{k>=0} (1 + x^(5*k*(k+1)/2+1)).

%C Number of partitions of n into distinct centered pentagonal numbers (A005891).

%H Alois P. Heinz, <a href="/A281083/b281083.txt">Table of n, a(n) for n = 0..20000</a> (first 1001 terms from G. C. Greubel)

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

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>

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

%F G.f.: Product_{k>=0} (1 + x^(5*k*(k+1)/2+1)).

%e a(82) = 2 because we have [76, 6] and [51, 31].

%t nmax = 105; CoefficientList[Series[Product[1 + x^(5 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A005891, A218380, A280952, A281081, A281082, A281084.

%K nonn,changed

%O 0,83

%A _Ilya Gutkovskiy_, Jan 14 2017