A172110 a(n) is the number of ordered partitions of {1, 1, 1, 2, 3, ..., n-2}.

0, 0, 4, 20, 132, 1076, 10404, 116180, 1469892, 20766836, 323924964, 5527326740, 102396386052, 2046350191796, 43876822764324, 1004631156809300, 24463049576172612, 631213045618035956, 17203155473859536484

Offset: 1

Author

Martin Griffiths, Jan 25 2010

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..420
• 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

Row sums of A172107.

Programs

Magma

[0,0] cat [(&+[ (&+[Binomial(k,j)*Binomial(j+2,3)*(-1)^(k-j)*j^(n-3): j in [0..k]]): k in [1..n]]): n in [3..25]]; // G. C. Greubel, Apr 15 2022

Mathematica

f[r_, n_]:= f[r, n]= If[n<3, 0, Sum[Sum[Binomial[m, l]Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l, m}], {m, n}]]; Table[f[3, n], {n, 25}]

Sage

[0,0]+[sum(sum(binomial(k,j)*binomial(j+2,3)*(-1)^(k+j)*j^(n-3) for j in (0..k)) for k in (1..n)) for n in (3..25)] # G. C. Greubel, Apr 15 2022

Formula

T_3(n) = Sum_{m=1..n} Sum_{j=0..m} binomial(m,j)*binomial(j+2,3)*(-1)^(m-j)*j^(n-3) for n > 2 with T_3(1) = T_3(2) = 0.

a(n) ~ n! / (12 * log(2)^(n+1)). - Vaclav Kotesovec, Apr 15 2022