A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

1, 1, 2, 9, 70, 805, 12480, 245847, 5909338, 168310515, 5556486450, 209003251240, 8835266400450, 415094928861530, 21473740362658640, 1213683089969940075, 74446121738526773490, 4927385997649620215895, 350145746700442604768346

Offset: 0

Author

Paul D. Hanna, May 09 2010

Keywords

• nonn

Examples

			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 +...
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 +...
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:
  1;
  1, 1;
  1, 2,  2,  2,  2;
  1, 3,  5,  7,  9,  9,   9,   9,   9,   9;
  1, 4,  9, 16, 25, 34,  43,  52,  61,  70,  70,  70,  70,  70,  70,  70,  70;
  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;
  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..356

Crossrefs

Cf. A107877, A177447, A177448, A177449, A209440.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial(1+ j^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 2022

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)}

Formula

G.f.: Sum_{n>=0} a(n)*x^n*(1-x)^(n^2) = 1/(1-x).

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.

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.