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Search: id:A051231
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| A051231 |
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Generalized Stirling number triangle of first kind. |
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+0
3
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| 1, -9, 1, 162, -27, 1, -4374, 891, -54, 1, 157464, -36450, 2835, -90, 1, -7085880, 1797714, -164025, 6885, -135, 1, 382637520, -104162436, 10655064, -535815, 14175, -189, 1, -24106163760, 6944870988, -775431468, 44411409, -1428840, 26082, -252, 1
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n,m)= R_n^m(a=0,b=9) in the notation of the given reference.
a(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n) = product(x-9*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
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REFERENCES
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Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
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FORMULA
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a(n, m) = a(n-1, m-1) - 9*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((ln(1+9*x))/9)^m)/m!.
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EXAMPLE
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{1}; {-9,1}; {162,-27,1}; {-4374,891,-54,1}; ... E(3,x) = 162*x-27*x^2+x^3.
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CROSSREFS
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First (m=1) column sequence is A051232(n-1). Row sums (signed triangle): A049211(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A045756(n). Cf. A051187 (b=8 triangle).
Sequence in context: A051380 A136238 A113394 this_sequence A046761 A019876 A155696
Adjacent sequences: A051228 A051229 A051230 this_sequence A051232 A051233 A051234
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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