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A140749
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Triangle c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.
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6
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1, -1, 1, 1, -1, 1, -1, 11, -3, 1, 1, -5, 7, -2, 1, -1, 137, -15, 17, -5, 1, 1, -7, 29, -7, 25, -3, 1, -1, 363, -469, 967, -35, 23, -7, 1, 1, -761, 29531, -89, 1069, -9, 91, -4, 1, -1, 7129, -1303, 4523, -285, 3013, -105, 29, -9, 1, 1, -671, 16103, -7645, 31063, -781, 4781, -55, 12, -5, 1
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OFFSET
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0,8
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COMMENTS
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The polynomials P(n,x) are defined in A129891: P(0,x)=1 and
P(n,x) = (-1)^n/(n+1) + x* Sum_{i=0..n-1) (-1)^i*P(n-1-i,x)/(i+1) = Sum_{k=0..n} binomial(n,k)*x^k.
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REFERENCES
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Paul Curtz, Gazette des Mathématiciens, 1992, 52, p. 44.
Paul Curtz, Intégration Numérique .. Note 12 du Centre de Calcul Scientifique de l'Armement, Arcueil, 1969. Now in 35170, Bruz.
P. Flajolet, X. Gourdon, and B. Salvy, Sur une famille de polynômes issus de l'analyse numérique, Gazette des Mathématiciens, 1993, 55, pp. 67-78.
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LINKS
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FORMULA
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(n+1)*c(n,k) = (n+1-k)*c(n-1,k) - n*c(n-1, k-1). [Edgard Bavencoffe in 1992]
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EXAMPLE
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The polynomials, for n =0,1,2, ..., are
P(0, x) = 1;
P(1, x) = -1/2 + x;
P(2, x) = 1/3 - x + x^2;
P(3, x) = -1/4 + 11/12*x - 3/2*x^2 + x^3;
P(4, x) = 1/5 - 5/6*x + 7/4*x^2 - 2*x^3 + x^4;
P(5, x) = -1/6 + 137/180*x - 15/8*x^2 + 17/6*x^3 - 5/2*x^4 + x^5;
and the coefficients are
1;
-1/2, 1;
1/3, -1, 1;
-1/4, 11/12, -3/2, 1;
1/5, -5/6, 7/4, -2, 1;
-1/6, 137/180, -15/8, 17/6, -5/2, 1;
1/7, -7/10, 29/15, -7/2, 25/6, -3, 1;.
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MAPLE
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P := proc(n, x) option remember ; if n =0 then 1; else (-1)^n/(n+1)+x*add( (-1)^i/(i+1)*procname(n-1-i, x), i=0..n-1) ; expand(%) ; fi; end:
A140749 := proc(n, k) p := P(n, x) ; numer(coeftayl(p, x=0, k)) ; end: seq(seq(A140749(n, k), k=0..n), n=0..13) ; # R. J. Mathar, Aug 24 2009
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MATHEMATICA
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p[0] = 1; p[n_] := p[n] = (-1)^n/(n+1) + x*Sum[(-1)^k*p[n-1-k] / (k+1), {k, 0, n-1}];
Numerator[ Flatten[ Table[ CoefficientList[p[n], x], {n, 0, 11}]]][[1 ;; 69]] (* Jean-François Alcover, Jun 17 2011 *)
Table[Numerator[(k+1)!*StirlingS1[n+1, k+1]/(n+1)!], {n, 0, 12}, {k, 0, n} ]//Flatten (* G. C. Greubel, Oct 24 2023 *)
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PROG
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(Magma) [Numerator(Factorial(k+1)*StirlingFirst(n+1, k+1)/Factorial(n+1) ): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023
(SageMath)
def A048594(n, k): return (-1)^(n-k)*numerator(factorial(k+1)* stirling_number1(n+1, k+1)/factorial(n+1))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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