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A337798 Number of partitions of the n-th n-gonal pyramidal number into distinct n-gonal pyramidal numbers.

%I A337798 #17 Feb 16 2025 08:34:00
%S A337798 1,1,1,1,1,1,1,1,1,2,1,1,3,1,1,2,4,5,4,5,7,11,9,4,12,12,24,23,42,59,
%T A337798 64,58,124,206,212,168,377,539,703,873,1122,1505,1943,2724,4100,4513,
%U A337798 6090,7138,12079,16584,20240,27162,35874,52622,69817,88059,115628,152756,219538,240200,358733,480674
%N A337798 Number of partitions of the n-th n-gonal pyramidal number into distinct n-gonal pyramidal numbers.
%H A337798 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>
%H A337798 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A337798 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F A337798 a(n) = [x^p(n,n)] Product_{k=1..n} (1 + x^p(n,k)), where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.
%e A337798 a(9) = 2 because the ninth 9-gonal pyramidal number is 885 and we have [885] and [420, 266, 155, 34, 10].
%p A337798 p:= (n,k) ->  k * (k + 1) * (k * (n - 2) - n + 5) / 6:
%p A337798 f:= proc(n) local k, P;
%p A337798   P:= mul(1+x^p(n,k),k=1..n);
%p A337798   coeff(P,x,p(n,n));
%p A337798 end proc:
%p A337798 map(f, [$0..80]); # _Robert Israel_, Sep 23 2020
%o A337798 (PARI) default(parisizemax, 2^31);
%o A337798 p(n,k) = k*(k + 1)*(k*(n-2) - n + 5)/6;
%o A337798 a(n) = my(f=1+x*O(x^p(n,n))); for(k=1, n, f*=1+x^p(n,k)); polcoeff(f, p(n,n)); \\ _Jinyuan Wang_, Dec 21 2021
%Y A337798 Cf. A006484, A288126, A298857, A337763, A337797, A337799.
%K A337798 nonn
%O A337798 0,10
%A A337798 _Ilya Gutkovskiy_, Sep 22 2020
%E A337798 More terms from _Robert Israel_, Sep 23 2020