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 A174128 Irregular triangle read by rows: row n (n > 0) is the expansion of Sum_{m=1..n} A001263(n,m)*x^(m - 1)*(1 - x)^(n - m). 3
 1, 1, 1, 1, -1, 1, 3, -3, 1, 6, -4, -4, 2, 1, 10, 0, -20, 10, 1, 15, 15, -55, 15, 15, -5, 1, 21, 49, -105, -35, 105, -35, 1, 28, 112, -140, -266, 364, -56, -56, 14, 1, 36, 216, -84, -882, 756, 336, -504, 126, 1, 45, 375, 210, -2100, 672, 2520, -2100, 210, 210, -42 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Row n gives the coefficients in the expansion of (1/x)*(1 - x)^n*N(n,x/(1 - x)), where N(n,x) is the n-th row polynomial for the triangle of Narayana numbers A001263. LINKS Michael Albert, Cheyne Homberger, Jay Pantone, Equipopularity Classes in the Separable Permutations, arXiv:1410.7312 [math.CO], (27-October-2014); see p. 13. Wikipedia, Hypergeometric function FORMULA The n-th row of the triangle is generated by the coefficients of (1 - x)^(n - 1)*F(-n, 1 - n; 2; x/(1 - x)), where F(a, b ; c; z) is the ordinary hypergeometric function. G.f.: ((1 - y - sqrt(1 - 2*y + ((1 - 2*x)*y)^2))/(2*(1 - x)*x*y). - Franck Maminirina Ramaharo, Oct 23 2018 EXAMPLE Triangle begins     1;     1;     1,  1,  -1;     1,  3,  -3;     1,  6,  -4,   -4,    2;     1, 10,   0,  -20,   10;     1, 15,  15,  -55,   15,  15,  -5;     1, 21,  49, -105,  -35, 105, -35;     1, 28, 112, -140, -266, 364, -56,  -56,  14;     1, 36, 216,  -84, -882, 756, 336, -504, 126;     ... MATHEMATICA p[x_, n_] = Sum[(Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m)*x^m*(1 - x)^(n - m), {m, 1, n}]/x; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}]; Flatten[%] CROSSREFS Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123202, A123217, A123221, A141720, A144387, A144400. Sequence in context: A016454 A065227 A208539 * A131070 A295290 A165202 Adjacent sequences:  A174125 A174126 A174127 * A174129 A174130 A174131 KEYWORD sign,tabf AUTHOR Roger L. Bagula, Mar 09 2010 EXTENSIONS Edited and new name by Joerg Arndt, Oct 28 2014 Comments and formula clarified by Franck Maminirina Ramaharo, Oct 23 2018 STATUS approved

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Last modified September 20 00:52 EDT 2020. Contains 337228 sequences. (Running on oeis4.)