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A039744 Number of ways n*(n-1) can be partitioned into the sum of 2*(n-1) integers in the range 0..n. 5
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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A045612 A103940 A162543 * A344262 A352985 A319121
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

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Last modified February 24 17:27 EST 2024. Contains 370307 sequences. (Running on oeis4.)