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 A138022 Triangular array read by rows: e.g.f. sqrt(1-z^2)*exp(x*z)/(1+z). 1
 1, -1, 1, 1, -2, 1, -3, 3, -3, 1, 9, -12, 6, -4, 1, -45, 45, -30, 10, -5, 1, 225, -270, 135, -60, 15, -6, 1, -1575, 1575, -945, 315, -105, 21, -7, 1, 11025, -12600, 6300, -2520, 630, -168, 28, -8, 1, -99225, 99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1, 893025, -992250, 496125, -189000, 47250, -11340, 1890, -360, 45, -10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS Table of n, a(n) for n=1..66. FORMULA The unsigned version has the e.g.f. exp(x*z)/sqrt((1-z)/(1+z)). - Peter Luschny, Aug 21 2014 T(n+3,k+1) = T(n+2,k) - T(n+2,k+1) + (n+1)*(n+2)*(T(n+1,k+1)-T(n,k)) with T(n,n) = 1, T(n,n-1) = -n, T(n+2,0) = T(n+1,0) + (n^2+n)*T(n,0). - Robert Israel, Aug 21 2014 T(n, k) = 1 if n = k, otherwise (-1)^(n+k)*(n-k)!*Sum_{i = 1..n-k} (Sum_{j = i..n-k} 2^(j-i)*Stirling1(j, i)*binomial(n-k-1, j-1)/j!)*binomial(n, k). - Detlef Meya, Jan 16 2024 EXAMPLE Triangle starts: 1; -1, 1; 1, -2, 1; -3, 3, -3, 1; 9, -12, 6, -4, 1; -45, 45, -30, 10, -5, 1; 225, -270, 135, -60, 15, -6, 1; -1575, 1575, -945, 315, -105, 21, -7, 1; 11025, -12600, 6300, -2520, 630, -168, 28, -8, 1; -99225, 99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1; ... MAPLE g := sqrt(1-z^2)*exp(x*z)/(1+z); gser := n -> series(g, z, n+2): seq(print(seq(coeff(n!*coeff(gser(n), z, n), x, i), i=0..n)), n=0..10); # Peter Luschny, Aug 21 2014 MATHEMATICA max=10; g=Exp[x*z]*Sqrt[(1-z)/(1+z)]; gser=Series[g, {z, 0, max}]; p[n_]:=n!*Coefficient[gser, z, n]; T[n_, k_]:=Coefficient[p[n], x, k]; Flatten[Table[T[n, k], {n, 0, max}, {k, 0, n}]] T[n_, k_] := If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 16 2024 *) CROSSREFS Cf. A130915 (row sums). Sequence in context: A227532 A152534 A136018 * A113278 A132382 A366627 Adjacent sequences: A138019 A138020 A138021 * A138023 A138024 A138025 KEYWORD tabl,sign AUTHOR Roger L. Bagula, May 01 2008 EXTENSIONS Edited by Peter Luschny and Joerg Arndt, Aug 21 2014 STATUS approved

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Last modified April 20 03:59 EDT 2024. Contains 371798 sequences. (Running on oeis4.)