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 A049460 Generalized Stirling number triangle of first kind. 8
 1, -5, 1, 30, -11, 1, -210, 107, -18, 1, 1680, -1066, 251, -26, 1, -15120, 11274, -3325, 485, -35, 1, 151200, -127860, 44524, -8175, 835, -45, 1, -1663200, 1557660, -617624, 134449, -17360, 1330, -56, 1, 19958400, -20355120, 8969148, -2231012, 342769, -33320, 2002, -68, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n,m)= ^5P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(5+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(5*t),exp(t)-1). LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962). FORMULA a(n, m)= a(n-1, m-1) - (n+4)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n (zipWith (-) ([0] ++ row) \$ map (* i) (row ++ [0]), i + 1)) ([1], 5) -- Reinhard Zumkeller, Mar 11 2014 CROSSREFS Unsigned column sequences are: A001720-A001724. Row sums (signed triangle): A001715(n+3)*(-1)^n. Row sums (unsigned triangle): A001725(n+5). Cf. A000035 A084938. Sequence in context: A144890 A144891 A135892 * A145926 A062140 A144355 Adjacent sequences: A049457 A049458 A049459 * A049461 A049462 A049463 KEYWORD sign,easy,tabl AUTHOR Wolfdieter Lang EXTENSIONS Second formula corrected by Philippe Deléham, Nov 10 2008 STATUS approved

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Last modified August 9 10:37 EDT 2024. Contains 375040 sequences. (Running on oeis4.)