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%I #13 Aug 28 2019 23:02:33
%S 1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,6,6,7,7,7,7,8,8,9,9,
%T 9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,18,18,19,19,20,20,22,
%U 22,23,23,25,25,27,27,28,28,30,30,32,32,34,34,36,36,38,38,40,40,42,42,45,45,47,47,49,49,52,52,54,54,57
%N Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)^2*(k+2)/12)).
%C Number of partitions of n into nonzero 4-dimensional pyramidal numbers (A002415).
%H Robert Israel, <a href="/A290792/b290792.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)^2*(k+2)/12)).
%e a(12) = 3 because we have [6, 6], [6, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
%p N:= 100: # for a(0)..a(N)
%p P:= 1:
%p for k from 1 do
%p e:= k*(k+1)^2*(k+2)/12;
%p if e > N then break fi;
%p P:= P/(1-x^e);
%p od:
%p S:= series(P,x,N+1):
%p [seq](coeff(S,x,n),n=0..N); # _Robert Israel_, Aug 28 2019
%t nmax = 90; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)^2 (k + 2)/12)), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000330, A001156, A002415, A279220, A290573.
%K nonn
%O 0,7
%A _Ilya Gutkovskiy_, Aug 15 2017
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