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

0,3

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)

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

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
      <n, 0, b(n, i-1, t)+`if`(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 *)

Prog

(Sage) def a039744(n): return gaussian_binomial(3*n-2, n)[n*(n-1)] # Max Alekseyev, Jun 16 2023
(PARI) a039744(n) = polcoef(matpascal(3*n-1, x)[3*n-1, n+1], n*(n-1)); \\ Max Alekseyev, Jun 16 2023

Crossrefs

Cf. A007878, A118113, A362967.

Sequence in context: A371494 A103940 A162543 * A344262 A352985 A319121

Adjacent sequences: A039741 A039742 A039743 * A039745 A039746 A039747

Keyword

nonn

Author

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

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

Status

approved