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 A051379 Generalized Stirling number triangle of first kind. 10
 1, -8, 1, 72, -17, 1, -720, 242, -27, 1, 7920, -3382, 539, -38, 1, -95040, 48504, -9850, 995, -50, 1, 1235520, -725592, 176554, -22785, 1645, -63, 1, -17297280, 11393808, -3197348, 495544, -45815, 2527, -77, 1, 259459200, -188204400, 59354028, -10630508, 1182769, -83720, 3682, -92, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n,m)= ^8P_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-(8+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(8*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+7)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n (zipWith (-) ([0] ++ row) \$ map (* i) (row ++ [0]), i + 1)) ([1], 8) -- Reinhard Zumkeller, Mar 12 2014 CROSSREFS The first (m=0) column sequence is: A049388. Row sums (signed triangle): A001730(n+6)*(-1)^n. Row sums (unsigned triangle): A049389(n). Cf. A000035 A084938. Sequence in context: A038279 A075503 A260040 * A143499 A114152 A347111 Adjacent sequences: A051376 A051377 A051378 * A051380 A051381 A051382 KEYWORD sign,easy,tabl AUTHOR Wolfdieter Lang EXTENSIONS Typo fixed in data by Reinhard Zumkeller, Mar 12 2014 STATUS approved

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Last modified November 30 02:35 EST 2023. Contains 367452 sequences. (Running on oeis4.)