A172111 a(n) is the number of ordered partitions of {1, 1, 1, 1, 2, 3, ..., n-3}.

0, 0, 0, 8, 48, 368, 3408, 36848, 454608, 6294128, 96556368, 1624775408, 29744591568, 588384837488, 12503968334928, 284065406275568, 6869235761650128, 176150548586638448, 4774198652678411088

Offset: 1

Author

Martin Griffiths, Jan 25 2010

Keywords

• nonn

Comments

a(n) is T_4(n) in the Griffiths and Mezo reference. - G. C. Greubel, Apr 15 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 1..400
• M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.

Crossrefs

This gives the row sums of A172108.

Cf. A172109, A172110.

Programs

Magma

[0,0,0] cat [(&+[ (&+[Binomial(k,j)*Binomial(j+3,4)*(-1)^(k-j)*j^(n-4): j in [0..k]]): k in [1..n]]): n in [4..25]]; // G. C. Greubel, Apr 15 2022

Mathematica

f[r_, n_]:= If[n<4, 0, Sum[Sum[Binomial[m,l]Binomial[l+r-1,r](-1)^(m-l)l^(n-r), {l,m}], {m, n}]]; Table[f[4, n], {n,25}]

Sage

[0,0,0]+[sum(sum(binomial(k,j)*binomial(j+3,4)*(-1)^(k+j)*j^(n-4) for j in (0..k)) for k in (1..n)) for n in (4..25)] # G. C. Greubel, Apr 15 2022

Formula

a(n) = Sum_{m=1..n} Sum_{j=0..m} binomial(m,j)*binomial(j+3,4)*(-1)^(m-j)*j^(n-4), for n>=4, with a(n) = 0 for n < 4.

a(n) ~ n! / (48 * log(2)^(n+1)). - Vaclav Kotesovec, Apr 15 2022