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A078739
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Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).
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16
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1, 2, 4, 1, 4, 32, 38, 12, 1, 8, 208, 652, 576, 188, 24, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 32, 7744, 116656, 412800, 540080, 322848, 98292, 16000, 1390, 60, 1, 64, 46592, 1446368, 9196992, 20447056, 20453376, 10564304, 3047520, 511392, 50400
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The g.f. for column k=2*K is (x^K)*pe(K,x)*d(k,x) and for k=2*K+1 it is (x^K)*po(K,x)*2*(K+1)*K*d(k,x), K>= 1, with d(k,x) := 1/product(1-p*(p-1)*x,p=2..k) and the row polynomials pe(n,x) := sum(A089275(n,m)*x^m,m=0..n-1) and po(n,x) := sum(A089276(n,m)*x^m,m=0..n-1). - Wolfdieter Lang, Nov 07 2003
The formula for the k-th column sequence is given in A089511.
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REFERENCES
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.
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LINKS
| P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
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FORMULA
| a(n, k)=sum(binomial(k-2+p, p)*A008279(2, p)*a(n-1, k-2+p), p=0..2) if 2 <= k <= 2*n for n>=1, a(1, 2)=1; else 0. Here A008279(2, p) gives the third row (k=2) of the augmented falling factorial triangle: [1, 2, 2] for p=0, 1, 2. From eq.(21) with r=2 of the Blasiak et al. paper.
a(n, k)= (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*A008279(p, 2)^n, p=2..k) for 2 <= k <= 2*n, n>=1. From eq.(19) with r=2 of the Blasiak et al. paper.
a(n, k)= sum(A071951(n, j)*A089503(j, 2*j-k+1), j=ceiling(k/2)..min(n, k-1)), 1<=n, 2<=k<=2n; relation to Legendre-Stirling triangle. Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003
a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011
E^n = sum_{k=2}^(2n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^2d^2/dx^2.
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EXAMPLE
| 1; 2,4,1; 4,32,38,12,1; 8,208,625,576,188,24,1; ...
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MAPLE
| # Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A078739 := proc(n, k) local r;
add((-1)^(n-r)*binomial(n, r)*combinat[stirling2](n+r, k), r=0..n) end:
# Displays the truncated triangle from the definition:
seq(print(seq(A078739(n, k), k=2..2*n)), n=1..6); # - Peter Luschny, Mar 25 2011
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CROSSREFS
| Row sums give A020556. Triangle S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{3, 2} = A078740, S_{3, 3} = A078741.
The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.
Cf. A071951 (Legendre-Stirling triangle).
Cf. A122193, A055203.
Sequence in context: A181332 A204021 A126126 * A004597 A077623 A132042
Adjacent sequences: A078736 A078737 A078738 * A078740 A078741 A078742
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KEYWORD
| nonn,tabf,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 21 2002
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EXTENSIONS
| More terms from Wolfdieter Lang, Nov 07 2003
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