A128552 Column 2 of triangle A128545; a(n) is the coefficient of q^(2n+4) in the central q-binomial coefficient [2n+4,n+2].

1, 3, 8, 18, 39, 75, 141, 251, 433, 724, 1185, 1892, 2972, 4588, 6981, 10480, 15553, 22821, 33164, 47746, 68163, 96542, 135747, 189550, 262997, 362691, 497339, 678300, 920417, 1242898, 1670688, 2235880, 2979809, 3955422, 5230471, 6891234

Offset: 0

Author

Paul D. Hanna, Mar 10 2007

Keywords

• nonn

Comments

Column 1 of triangle A128552 equals the partitions of n (A000041).

a(n) is the number of partitions of the integer 2n+4 into at most n+2 summands each of which is at most n+2. - Geoffrey Critzer, Sep 27 2013

Examples

			a(2) = 8 because we have: 4+4 = 4+3+1 = 4+2+2 = 4+2+1+1 = 3+3+2 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2. - _Geoffrey Critzer_, Sep 27 2013
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45.
• Shishuo Fu and James Sellers, Enumeration of the degree sequences of line-Hamiltonian multigraphs, INTEGERS 12 (2012), #A24. - From _N. J. A. Sloane_, Feb 04 2013

Crossrefs

Cf. A128545; A128553, A128554.

Programs

Maple

with(combinat): p:= numbpart:
s:= proc(n) s(n):= p(n) +`if`(n>0, s(n-1), 0) end:
a:= n-> p(2*n+4) -2*s(n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 27 2013

Mathematica

Table[nn=2n;Coefficient[Series[Product[(1-x^(n+i))/(1-x^i),{i,1,n}],{x,0,nn}],x^(2n)],{n,1,37}] (* Geoffrey Critzer, Sep 27 2013 *)

PARI

{a(n)=polcoeff(prod(j=n+3,2*n+4,1-q^j)/prod(j=1,n+2,1-q^j),2*n+4,q)}

PARI

{a(n)=numbpart(2*n+4)-2*sum(k=0,n+1,numbpart(k))} \\ Paul D. Hanna, Feb 06 2013

Formula

a(n) = A000041(2*n+4) - 2*Sum_{k=0..n+1} A000041(k), where A000041(n) = number of partitions of n, due to a formula given in the Fu and Sellers paper. - Paul D. Hanna, Feb 06 2013