A177447 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2) = 1+x.

1, 1, 1, 3, 18, 172, 2313, 40626, 887326, 23282964, 715540140, 25259729071, 1008721104654, 45008479039824, 2221170817590696, 120209722115431950, 7083266027910364710, 451620678137942740132, 30990400538494184551692, 2277988537997377457967690, 178626191260072536476398000

Offset: 0

Author

Paul D. Hanna, May 09 2010

Keywords

• nonn

Comments

Column 1 of triangle A215241.

Examples

			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 +...
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:
  1;
  1, 1,  1;
  1, 2,  3,  3,  3,   3,   3;
  1, 3,  6,  9, 12,  15,  18,  18,  18,  18,  18,  18,  18;
  1, 4, 10, 19, 31,  46,  64,  82, 100, 118, 136, 154, 172,  172,  172,  172,  172,  172,  172,  172,  172;
  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; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A215241, A215242, A215243, A133316, A177448, A177449, A177450.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial(1 +j*(j-1), n-j), j=0..n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 10 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^2)),n)}

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],", "))}

Formula

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.

Generating functions:

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

(2) 1/(1-x) = Sum_{n>=0} a(n) * x^n * (1-x)^(n*(n-1)). - Paul D. Hanna, Apr 04 2022