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Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)/2+1)).
8

%I #11 May 08 2017 00:30:30

%S 1,1,1,1,2,2,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11,11,12,13,15,15,16,17,19,

%T 20,22,24,26,27,29,31,33,34,37,40,43,45,48,51,54,56,60,63,67,70,76,80,

%U 84,87,93,97,102,106,113,118,125,130,138,143,151,157,166,172,181,189,200,207,217,225,237,245,257,267,280

%N Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)/2+1)).

%C Number of partitions of n into centered triangular numbers (A005448).

%H Robert Israel, <a href="/A280950/b280950.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular 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/(1 - x^(3*k*(k+1)/2+1)).

%e a(8) = 3 because we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

%p N:= 100:

%p kmax:= floor((sqrt(24*N-15)-3)/6):

%p S:= series(mul(1/(1-x^(3*k*(k+1)/2+1)),k=0..kmax),x,N+1):

%p seq(coeff(S,x,j),j=0..N); # _Robert Israel_, Jan 25 2017

%t nmax = 78; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A005448, A007294, A068980, A280951, A280952, A280953.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Jan 11 2017