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A071951
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Triangle of Legendre-Stirling numbers of the second kind T(n,j), n>=1, 1<=j<=n, read by rows.
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17
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1, 2, 1, 4, 8, 1, 8, 52, 20, 1, 16, 320, 292, 40, 1, 32, 1936, 3824, 1092, 70, 1, 64, 11648, 47824, 25664, 3192, 112, 1, 128, 69952, 585536, 561104, 121424, 7896, 168, 1, 256, 419840, 7096384, 11807616, 4203824, 453056, 17304, 240, 1, 512
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
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LINKS
| G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers
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FORMULA
| T(n, j) = sum_{r=1..j} (-1)^{r+j}(2r+1)(r^2+r)^n/((r+j+1)!(j-r)!).
G.f. for j-th column (without leading zeros): 1/product(1-r*(r+1)*x, r=1..j), j>=1. From eq.(4.5) of the Everitt et al. paper.
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EXAMPLE
| 1; 2,1; 4,8,1; 8,52,20,1; 16,320,292,40,1; ...
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MATHEMATICA
| Flatten[ Table[ Sum[(-1)^{r + j}(2r + 1)(r^2 + r)^n/((r + j + 1)!(j - r)!), {r, 1, j}], {n, 1, 10}, {j, 1, n}]]
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CROSSREFS
| Diagonals give A007290, A000079, A016129, A016309.
The column sequences are A000079 (powers of 2), A016129, A016309, A071952, A089274, A089277.
Cf. A089278, A089500.
Sequence in context: A011234 A161381 A128412 * A160323 A128411 A164614
Adjacent sequences: A071948 A071949 A071950 * A071952 A071953 A071954
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 16 2002
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