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A193685
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5-restricted Stirling numbers of the second kind.
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13
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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
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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Vincenzo Librandi, Rows n = 0..100, flattened
Broder Andrei Z., The r-Stirling numbers, Discrete Math. 49, 241-259 (1984)
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FORMULA
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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., 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).
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EXAMPLE
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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.
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MATHEMATICA
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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}] ] (* From Jean-François Alcover, Dec 02 2011, after Wolfdieter Lang *)
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CROSSREFS
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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
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang, Oct 06 2011
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STATUS
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approved
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