|
%I #9 Jul 10 2022 08:44:42
%S 1,1,2,9,70,805,12480,245847,5909338,168310515,5556486450,
%T 209003251240,8835266400450,415094928861530,21473740362658640,
%U 1213683089969940075,74446121738526773490,4927385997649620215895,350145746700442604768346
%N G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.
%H Alois P. Heinz, <a href="/A177450/b177450.txt">Table of n, a(n) for n = 0..356</a>
%F G.f.: Sum_{n>=0} a(n)*x^n*(1-x)^(n^2) = 1/(1-x).
%F G.f.: Sum_{n>=0} a(n)*x^n*C(-x)^(n^2+2n) = 1/C(-x) where C(x) is the Catalan function of A000108.
%F a(n) = number of subpartitions of partition consisting of the first n square numbers starting with zero for n>0; e.g., a(4) = subp([0,1,4,9]) = 70. See A115728 for the definition of subpartitions.
%e 1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^6 + 9*x^3/(1+x)^12 + 70*x^4/(1+x)^20 + 805*x^5/(1+x)^30 +...
%e 1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^4 + 9*x^3*(1-x)^9 + 70*x^4*(1-x)^16 + 805*x^5*(1-x)^25 +...
%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 2(n+1) times, starting with a '1' in row 0, as illustrated by:
%e 1;
%e 1, 1;
%e 1, 2, 2, 2, 2;
%e 1, 3, 5, 7, 9, 9, 9, 9, 9, 9;
%e 1, 4, 9, 16, 25, 34, 43, 52, 61, 70, 70, 70, 70, 70, 70, 70, 70;
%e 1, 5, 14, 30, 55, 89, 132, 184, 245, 315, 385, 455, 525, 595, 665, 735, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805;
%e ...
%p a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
%p *(-1)^(n-j)*binomial(1+ j^2, n-j), j=0..n-1))
%p end:
%p seq(a(n), n=0..19); # _Alois P. Heinz_, Jul 08 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*(k+1))),n)}
%Y Cf. A107877, A177447, A177448, A177449, A209440.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 09 2010
|
