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 A049458 Generalized Stirling number triangle of first kind. 11
 1, -3, 1, 12, -7, 1, -60, 47, -12, 1, 360, -342, 119, -18, 1, -2520, 2754, -1175, 245, -25, 1, 20160, -24552, 12154, -3135, 445, -33, 1, -181440, 241128, -133938, 40369, -7140, 742, -42, 1, 1814400, -2592720, 1580508, -537628 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n,m)= ^3P_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-(3+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 polynomials for (exp(3*t),exp(t)-1). See A143492 for the unsigned version of this array and A143495 for the inverse. - Peter Bala, Aug 25 2008 REFERENCES 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. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened FORMULA a(n, m)= a(n-1, m-1) - (n+2)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n seq((-1)^(n-k)*coeff(expand(pochhammer(x+3, n)), x, k), k=0..n): seq(print(A049458_row(n)), n=0..8); # Peter Luschny, May 16 2013 MATHEMATICA t[n_, k_] := (-1)^(n - k)*Coefficient[ Pochhammer[x + 3, n], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 17 2013, after Peter Luschny *) PROG (Haskell) a049458 n k = a049458_tabl !! n !! k a049458_row n = a049458_tabl !! n a049458_tabl = map fst \$ iterate (\(row, i) ->    (zipWith (-) ([0] ++ row) \$ map (* i) (row ++ [0]), i + 1)) ([1], 3) -- Reinhard Zumkeller, Mar 11 2014 CROSSREFS Unsigned column sequences are: A001710-A001714. Row sums (signed triangle): (n+1)!*(-1)^n. Row sums (unsigned triangle): A001715(n+3). Cf. A000035 A084938. Cf. A094645, A094646. A143492, A143495. - Peter Bala, Aug 25 2008 Sequence in context: A329433 A258245 A133366 * A143492 A243662 A062139 Adjacent sequences:  A049455 A049456 A049457 * A049459 A049460 A049461 KEYWORD sign,easy,tabl AUTHOR EXTENSIONS Second formula corrected by Philippe Deléham, Nov 09 2008 STATUS approved

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Last modified April 17 04:36 EDT 2021. Contains 343059 sequences. (Running on oeis4.)