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 A051339 Generalized Stirling number triangle of first kind. 10
 1, -7, 1, 56, -15, 1, -504, 191, -24, 1, 5040, -2414, 431, -34, 1, -55440, 31594, -7155, 805, -45, 1, 665280, -434568, 117454, -16815, 1345, -57, 1, -8648640, 6314664, -1961470, 336049, -34300, 2086, -70, 1, 121080960, -97053936, 33775244, -6666156, 816249, -63504, 3066, -84, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n,m)= ^7P_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-(7+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(7*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+6)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n    (zipWith (-) ([0] ++ row) \$ map (* i) (row ++ [0]), i + 1)) ([1], 7) -- Reinhard Zumkeller, Mar 11 2014 CROSSREFS The first (m=0) column sequence is A001730. Row sums (signed triangle): A001725(n+5)*(-1)^n. Row sums (unsigned triangle): A049388(n). Cf. A000035 A084938. Sequence in context: A075502 A052104 A144450 * A134141 A237111 A281620 Adjacent sequences:  A051336 A051337 A051338 * A051340 A051341 A051342 KEYWORD sign,easy,tabl AUTHOR STATUS approved

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Last modified May 27 17:53 EDT 2020. Contains 334664 sequences. (Running on oeis4.)