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 A178667 Irregular triangle: T(n,k) is the coefficient [x^k] of the series (-1)^n *(x-1)^(n+2) *sum_{j=0..infinity} x^j /Beta(n+1,2*j+1), k=0..1+n/2, where Beta() is the usual Gamma-function ratio. 1
 1, 1, 2, 6, 3, 18, 3, 4, 40, 20, 5, 75, 75, 5, 6, 126, 210, 42, 7, 196, 490, 196, 7, 8, 288, 1008, 672, 72, 9, 405, 1890, 1890, 405, 9, 10, 550, 3300, 4620, 1650, 110, 11, 726, 5445, 10164, 5445, 726, 11 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The even-indexed rows (at least if limited to k<=1+n/2) are left-right symmetric. LINKS G. C. Greubel, Rows n=0..100 of triangle, flattened EXAMPLE 1, 1; 2, 6; 3, 18, 3; 4, 40, 20; 5, 75, 75, 5; 6, 126, 210, 42; 7, 196, 490, 196, 7; 8, 288, 1008, 672, 72; 9, 405, 1890, 1890, 405, 9; 10, 550, 3300, 4620, 1650, 110; 11, 726, 5445, 10164, 5445, 726, 11; MAPLE A178667 := proc(n, k) (-1)^n*(x-1)^(n+2)*add(x^j/Beta(n+1, 2*j+1), j=0..n+1) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Feb 12 2013 MATHEMATICA p[x_, n_] = (-1)^n*(-1 + x)^(n + 2)*Sum[(1/Beta[n + 1, 2*k + 1])x^k, {k, 0, Infinity}]; Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]] CROSSREFS Cf. A036289 (Row sums). Sequence in context: A083169 A276817 A050125 * A281881 A206493 A359256 Adjacent sequences: A178664 A178665 A178666 * A178668 A178669 A178670 KEYWORD nonn,tabf AUTHOR Roger L. Bagula, Jun 02 2010 STATUS approved

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Last modified June 8 18:04 EDT 2023. Contains 363165 sequences. (Running on oeis4.)