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A193685 5-restricted Stirling numbers of the second kind. 13
1, 5, 1, 25, 11, 1, 125, 91, 18, 1, 625, 671, 217, 26, 1, 3125, 4651, 2190, 425, 35, 1, 15625, 31031, 19981, 5590, 740, 45, 1, 78125, 201811, 170898, 64701, 12250, 1190, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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

a(n,m), n>=m>=0, counts 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.

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(a(n,m)*(a*)^(5+m)*a^m,m=0..n), n>=0. See the Mikhailov papers, where a(n,m) = S(n+5,m+5,5).

  With a->D=d/dx and a*->x one has also

  (xD)^n x^5 = sum(a(n,m)*x^(5+m)*D^m,m=0..n),n>=0.

LINKS

Vincenzo Librandi, Rows n = 0..100, flattened

P. Bala, Generalized Dobinski formulas

Broder Andrei Z., The r-Stirling numbers, Discrete Math. 49, 241-259 (1984)

V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.

V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.

FORMULA

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

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

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

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

  1/product(1-(5+j)*x,j=0..m), 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(1-(5+j)*x,j=0..m. This works for every r-restricted Stirling2 triangle.)

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.

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

Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*sum {k = 0..inf} k*(4 + k)^(n-1)*x^(k-1)/k!. - Peter Bala, Jun 23 2014

EXAMPLE

n\m  0    1    2   3  4  5 ...

0    1

1    5    1

2   25   11    1

3  125   91   18   1

4  625  671  217  26  1

5 3125 4651 2190 425 35  1

...

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

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

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

Normal ordering (n=1): (xD)^1 x^5 =  sum(a(1,m)*x^(5+m)*D^m,m=0..1)= 5*x^5 + 1*x^6*D.

a(2,1) = sum(S1(5,5-j)*S2(7-j,6),j=0..1) = 1*21-10*1 =11.

MATHEMATICA

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-Fran├žois Alcover, Dec 02 2011, after Wolfdieter Lang *)

CROSSREFS

Cf. A048993, A143494, A143495, A143496.

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

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

Sequence in context: A077195 A038243 A218016 * A174358 A075500 A096645

Adjacent sequences:  A193682 A193683 A193684 * A193686 A193687 A193688

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 06 2011

STATUS

approved

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Last modified October 30 07:54 EDT 2014. Contains 248796 sequences.