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A280953 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

%I A280953 #12 Feb 16 2025 08:33:39
%S A280953 1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,5,6,6,7,7,7,7,7,8,
%T A280953 8,9,9,10,11,11,12,12,13,13,14,15,15,16,16,17,17,18,19,19,20,20,22,23,
%U A280953 24,25,25,27,27,29,30,31,32,32,34,34,36,37,38,39,40,43,44,46,47,48,50,51,54,55,57,58,59
%N A280953 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).
%C A280953 Number of partitions of n into centered hexagonal numbers (A003215).
%H A280953 Alois P. Heinz, <a href="/A280953/b280953.txt">Table of n, a(n) for n = 0..20000</a>
%H A280953 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A280953 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A280953 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexNumber.html">Hex Number</a>
%H A280953 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H A280953 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A280953 G.f.: Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).
%e A280953 a(14) = 3 because we have [7, 7], [7, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
%p A280953 h:= proc(n) option remember; `if`(n<0, 0, (t->
%p A280953       `if`(3*t*(t+1)+1>n, t-1, t))(1+h(n-1)))
%p A280953     end:
%p A280953 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
%p A280953       b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(3*i*(i+1)+1)))
%p A280953     end:
%p A280953 a:= n-> b(n, h(n)):
%p A280953 seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 28 2018
%t A280953 nmax = 86; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
%Y A280953 Cf. A003215, A278949, A279222, A280950, A280951, A280952.
%K A280953 nonn
%O A280953 0,8
%A A280953 _Ilya Gutkovskiy_, Jan 11 2017