A039744 Number of ways n*(n-1) can be partitioned into the sum of 2*(n-1) integers in the range 0..n.

1, 1, 2, 5, 18, 73, 338, 1656, 8512, 45207, 246448, 1371535, 7764392, 44585180, 259140928, 1521967986, 9020077206, 53885028921, 324176252022, 1962530559999, 11947926290396, 73108804084505, 449408984811980, 2774152288318052, 17190155366056138, 106894140685782646

Offset: 0

Author

Bill Daly (bill.daly(AT)tradition.co.uk)

Keywords

• nonn

Comments

An upper bound on A007878.

The indices of the odd terms appear to be A118113. - T. D. Noe, Dec 19 2006

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150 (terms n=1..65 from T. D. Noe)

Crossrefs

Cf. A007878, A118113, A362967.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
      n, 0, b(n-i, i, t-1))))
    end:
a:= n-> b(n*(n-1), n, 2*(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 15 2016

Mathematica

T[0,p_,m_]=1; T[k_,0,m_]=0; T[k_,p_,m_]:=T[k,p,m]=Sum[T[k+i,p-1,-i], {i,-m,-1}]; Table[T[n(n-1),2n-2,n], {n,40}] (* T. D. Noe, Dec 19 2006 *)

PARI

a039744(n) = polcoef(matpascal(3*n-1, x)[3*n-1, n+1], n*(n-1)); \\ Max Alekseyev, Jun 16 2023

Sage

def a039744(n): return gaussian_binomial(3*n-2, n)[n*(n-1)] # Max Alekseyev, Jun 16 2023

Formula

a(n) = T(n(n+1),2n-2,n), where T(k,p,m) is a recursive function that gives the number of partitions of k into p parts of 0..m. It is defined T(k,p,m) = sum_{i=1..m} T(k-i,p-1,i), with the boundary conditions T(0,p,m)=1 and T(k,0,m)=0 for all positive k, p and m. - T. D. Noe, Dec 19 2006

a(n) = coefficient of q^(n*(n-1)) in q-binomial(3*n-2, n). - Max Alekseyev, Jun 16 2023

a(n) ~ 3^(3*n - 3/2) / (Pi * n^2 * 2^(2*n - 1)). - Vaclav Kotesovec, Jun 17 2023

Extensions

Definition corrected by Jozsef Pelikan (pelikan(AT)cs.elte.hu), Dec 05 2006

More terms from T. D. Noe, Dec 19 2006

a(0)=1 prepended by Alois P. Heinz, May 15 2016