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%I #19 Jul 10 2022 09:00:14
%S 1,1,1,3,18,172,2313,40626,887326,23282964,715540140,25259729071,
%T 1008721104654,45008479039824,2221170817590696,120209722115431950,
%U 7083266027910364710,451620678137942740132,30990400538494184551692,2277988537997377457967690,178626191260072536476398000
%N G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2) = 1+x.
%C Column 1 of triangle A215241.
%H Paul D. Hanna, <a href="/A177447/b177447.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = number of subpartitions of the partition [0,0,2,6,12,...,(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.
%F Generating functions:
%F (1) 1 + x = Sum_{n>=0} a(n) * x^n / (1+x)^(n^2).
%F (2) 1/(1-x) = Sum_{n>=0} a(n) * x^n * (1-x)^(n*(n-1)). - _Paul D. Hanna_, Apr 04 2022
%e 1+x = 1 + 1*x/(1+x) + 1*x^2/(1+x)^4 + 3*x^3/(1+x)^9 + 18*x^4/(1+x)^16 + 172*x^5/(1+x)^25 + 2313*x^6/(1+x)^36 +...
%e Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 2n+1 times, starting with a '1' in row 0, as illustrated by:
%e 1;
%e 1, 1, 1;
%e 1, 2, 3, 3, 3, 3, 3;
%e 1, 3, 6, 9, 12, 15, 18, 18, 18, 18, 18, 18, 18;
%e 1, 4, 10, 19, 31, 46, 64, 82, 100, 118, 136, 154, 172, 172, 172, 172, 172, 172, 172, 172, 172;
%e 1, 5, 15, 34, 65, 111, 175, 257, 357, 475, 611, 765, 937, 1109, 1281, 1453, 1625, 1797, 1969, 2141, 2313, 2313, 2313, 2313, 2313, 2313, 2313, 2313, 2313; ...
%p a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
%p *(-1)^(n-j)*binomial(1 +j*(j-1), n-j), j=0..n-1))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jul 10 2022
%o (PARI) {a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(k^2)),n)}
%o (PARI) {A=[1,1];for(i=1,40,A=concat(A,-Vec(sum(n=0,#A-1,A[n+1]*x^n/(1+x+x*O(x^#A))^(n^2)))[#A+1]));for(n=0,#A-1,print1(A[n+1],", "))}
%Y Cf. A215241, A215242, A215243, A133316, A177448, A177449, A177450.
%K nonn
%O 0,4
%A _Paul D. Hanna_, May 09 2010
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