A039744 - OEIS

%I #25 Nov 03 2023 11:57:28

%S 1,1,2,5,18,73,338,1656,8512,45207,246448,1371535,7764392,44585180,

%T 259140928,1521967986,9020077206,53885028921,324176252022,

%U 1962530559999,11947926290396,73108804084505,449408984811980,2774152288318052,17190155366056138,106894140685782646

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

%C An upper bound on A007878.

%C The indices of the odd terms appear to be A118113. - _T. D. Noe_, Dec 19 2006

%H Alois P. Heinz, <a href="/A039744/b039744.txt">Table of n, a(n) for n = 0..150</a> (terms n=1..65 from T. D. Noe)

%F 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

%F a(n) = coefficient of q^(n*(n-1)) in q-binomial(3*n-2, n). - _Max Alekseyev_, Jun 16 2023

%F a(n) ~ 3^(3*n - 3/2) / (Pi * n^2 * 2^(2*n - 1)). - _Vaclav Kotesovec_, Jun 17 2023

%p b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i

%p       <n, 0, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i, t-1))))

%p     end:

%p a:= n-> b(n*(n-1), n, 2*(n-1)):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, May 15 2016

%t 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 *)

%o (Sage) def a039744(n): return gaussian_binomial(3*n-2, n)[n*(n-1)] # _Max Alekseyev_, Jun 16 2023

%o (PARI) a039744(n) = polcoef(matpascal(3*n-1, x)[3*n-1, n+1], n*(n-1)); \\ _Max Alekseyev_, Jun 16 2023

%Y Cf. A007878, A118113, A362967.

%K nonn

%O 0,3

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

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

%E More terms from _T. D. Noe_, Dec 19 2006

%E a(0)=1 prepended by _Alois P. Heinz_, May 15 2016