A236310 Expansion of Sum_{k>=0} x^((k+1)^2)/(1-x)^k.

0, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 52, 62, 74, 89, 108, 132, 162, 199, 244, 298, 362, 437, 524, 625, 743, 882, 1047, 1244, 1480, 1763, 2102, 2507, 2989, 3560, 4233, 5022, 5943, 7015, 8261, 9709, 11393, 13354, 15641, 18312, 21435, 25089, 29365, 34367, 40213, 47036, 54985, 64227, 74950

Offset: 0

Author

Joerg Arndt, Apr 22 2014

Keywords

• nonn

Comments

a(n) is the number of compositions of n such that the first part is equal to the number of parts and all parts are greater than or equal to the first part. - John Tyler Rascoe, Feb 10 2024

Examples

			From _John Tyler Rascoe_, Feb 10 2024: (Start)
The compositions for n = 9..11 are:
9:  [3,3,3], [2,7];
10: [3,4,3], [3,3,4], [2,8];
11: [3,4,4], [3,3,5], [3,5,3], [2,9].
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A098131 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).

Cf. A219282 (g.f. Sum_{k>=0} x^(k*(k+1)/2)/(1-x)^k).

Cf. A063978 (g.f. Sum_{k>=0} x^(k^2)/(1-x)^(k+1)).

Programs

PARI

N=66; q='q+O('q^N);
gf=sum(n=0, N, q^((n+1)^2) / (1-q)^n );
concat([0],Vec(gf))

Formula

G.f.: Sum_{k>=0} x^((k+1)^2)/(1-x)^k.

G.f.: Sum_{k>0} A(x,k) where A(x,k) = (x^k)*(x^k/(1-x))^(k-1) is the g.f. for compositions of this kind with first part k. - John Tyler Rascoe, Feb 10 2024