A172109 a(n) is the number of ordered partitions of {1,1,2,3,...,n-1}.

0, 2, 8, 44, 308, 2612, 25988, 296564, 3816548, 54667412, 862440068, 14857100084, 277474957988, 5584100659412, 120462266974148, 2772968936479604, 67843210855558628, 1757952715142990612, 48093560991292628228

Offset: 1

Author

Martin Griffiths, Jan 25 2010

Keywords

• nonn

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

Row sums of A172106.

Cf. A005649. - R. J. Mathar, Jan 28 2010

Cf. A083410.

Programs

Magma

[(&+[Factorial(j+1)*StirlingSecond(n-1,j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Apr 14 2022

Mathematica

f[r_, n_]:= Sum[Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l, m}], {m, n}]; Join[{0}, Table[f[2, n], {n,2,30}]]

PARI

a(n) = sum(k=1, n-1, stirling(n-1,k,2)*(k+1)!); \\ Michel Marcus, Apr 14 2022

SageMath

[sum( factorial(j+1)*stirling_number2(n-1,j) for j in (1..n-1) ) for n in (1..30)] # G. C. Greubel, Apr 14 2022

Formula

For n>=2, T_2(n) = Sum_{m=1..n} Sum_{l=0..m} C(m,l)*C(l+1,2)*(-1)^(m-l)*l^(n-2).

G.f.: 1/G(0) -1 where G(k) = 1 - x*(k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013

G.f.: 1/Q(0) -1, where Q(k) = 1 - x*(3*k+2) - 2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013

a(n) = Sum_{k=1..n-1} Stirling2(n-1,k)*(k+1)!. - Karol A. Penson, Sep 04 2015

a(n) ~ n! / (4 * log(2)^(n+1)). - Vaclav Kotesovec, Apr 15 2022