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5-Stirling numbers of the second kind.
13

%I #50 Jan 29 2022 09:40:50

%S 1,5,1,25,11,1,125,91,18,1,625,671,217,26,1,3125,4651,2190,425,35,1,

%T 15625,31031,19981,5590,740,45,1,78125,201811,170898,64701,12250,1190,

%U 56,1,390625,1288991,1398097,688506,174951,24150,1806,68,1,1953125,8124571,11075670,6906145,2263065,416451,44016,2622,81,1,9765625,50700551,85654261,66324830,27273730,6427575,900627,75480,3675,95,1

%N 5-Stirling numbers of the second kind.

%C This is the lower triangular Sheffer matrix (exp(5*x),exp(x)-1). For Sheffer matrices see the W. Lang link under A006232 with references, and the rules for the conversion to the umbral notation of S. Roman's book.

%C The general case is Sheffer (exp(r*x),exp(x)-1), r=0,1,..., corresponding to r-Stirling2 numbers with row and column offsets 0. See the Broder link for r-Stirling2 numbers with offset [r,r].

%C a(n,m), n >= m >= 0, gives the number of partitions of the set {1.2....,n+5} into m+5 nonempty distinct subsets such that 1,2,3,4 and 5 belong to distinct subsets.

%C a(n,m) appears in the following normal ordering of Bose operators a and a* satisfying the Lie algebra [a,a*]=1: (a*a)^n (a*)^5 = Sum_{m=0..n} a(n,m)*(a*)^(5+m)*a^m, n >= 0. See the Mikhailov papers, where a(n,m) = S(n+5,m+5,5).

%C With a->D=d/dx and a*->x we also have

%C (xD)^n x^5 = Sum_{m=0..n} a(n,m)*x^(5+m)*D^m, n >= 0.

%H Vincenzo Librandi, <a href="/A193685/b193685.txt">Rows n = 0..100, flattened</a>

%H Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>

%H Andrei Z. Broder, <a href="http://infolab.stanford.edu/TR/CS-TR-82-949.html">The r-Stirling numbers</a>, Discrete Math. 49, 241-259 (1984)

%H A. Dzhumadildaev and D. Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]

%H Askar Dzhumadil’daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

%H V. V. Mikhailov, <a href="http://dx.doi.org/10.1088/0305-4470/16/16/019">Ordering of some boson operator functions</a>, J. Phys A: Math. Gen. 16 (1983) 3817-3827.

%H V. V. Mikhailov, <a href="http://dx.doi.org/10.1088/0305-4470/18/2/012">Normal ordering and generalised Stirling numbers</a>, J. Phys A: Math. Gen. 18 (1985) 231-235.

%F E.g.f. of row polynomials s(n,x):=Sum_{m=0..n} a(n,m)*x^m: exp(5*z + x(exp(z)-1)).

%F E.g.f. of column no. m (with leading zeros):

%F exp(5*x)*((exp(x)-1)^m)/m!, m >= 0 (Sheffer).

%F O.g.f. of column no. m (without leading zeros):

%F 1/Product_{j=0..m} (1-(5+j)*x), m >= 0. (Compute the first derivative of the column e.g.f. and compare its Laplace transform with the partial fraction decomposition of the o.g.f. x^(m-1)/Product_{j=0..m} (1-(5+j)*x). This works for every r-restricted Stirling2 triangle.)

%F Recurrence: a(n,m) = (5+m)*a(n-1,m) + a(n-1,m-1), a(0,0)=1, a(n,m)=0 if n < m, a(n,-1)=0.

%F a(n,m) = Sum_{j=0..min(5,n-m)} S1(5,5-j)*S2(n+5-j,m+5), n >= m >= 0, with S1 and S2 the Stirling1 and Stirling2 numbers A008275 and A048993, respectively (see the Mikailov papers).

%F Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(4+k)^(n-1)*x^(k-1)/k!. - _Peter Bala_, Jun 23 2014

%e n\m 0 1 2 3 4 5 ...

%e 0 1

%e 1 5 1

%e 2 25 11 1

%e 3 125 91 18 1

%e 4 625 671 217 26 1

%e 5 3125 4651 2190 425 35 1

%e ...

%e 5-restricted S2: a(1,0)=5 from 1,6|2|3|4|5, 2,6|1|3|4|5,

%e 3,6|1|2|4|5, 4,6|1|2|3|5 and 5,6|1|2|3|4.

%e Recurrence: a(4,2) = (5+2)*a(3,2)+ a(3,1) = 7*18 + 91 = 217.

%e Normal ordering (n=1): (xD)^1 x^5 = Sum_{m=0..1} a(1,m)*x^(5+m)*D^m = 5*x^5 + 1*x^6*D.

%e a(2,1) = Sum_{j=0..1} S1(5,5-j)*S2(7-j,6) = 1*21 - 10*1 = 11.

%t a[n_, m_] := Sum[ StirlingS1[5, 5-j]*StirlingS2[n+5-j, m+5], {j, 0, Min[5, n-m]}]; Flatten[ Table[ a[n, m], {n, 0, 10}, {m, 0, n}] ] (* _Jean-François Alcover_, Dec 02 2011, after _Wolfdieter Lang_ *)

%Y Cf. A048993, A143494, A143495, A143496.

%Y Cf. A196834 (row sums), A196835 (alternating row sums).

%Y Columns: A000351 (m=0), A005062 (m=1), A019757 (m=2), A028165 (m=3), ...

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Oct 06 2011