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 A178657 Irregular triangle: the coefficient [x^k] of the polynomial (1-x)^(2*n-1) * Sum_{s>=0} A001263(n+2*s,2*s+1)*x^s in row n >= 1 and column k >= 0. 1
 1, 1, 3, 1, 15, 15, 1, 1, 43, 161, 105, 10, 1, 96, 855, 1680, 855, 96, 1, 1, 185, 3191, 13387, 17655, 7623, 945, 21, 1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1, 1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36, 1, 808 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row sums are 1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, ... (see A151403, A052704). The sequence is the Narayana number analog of A034839. LINKS EXAMPLE 1; 1, 3; 1, 15, 15, 1; 1, 43, 161, 105, 10; 1, 96, 855, 1680, 855, 96, 1; 1, 185, 3191, 13387, 17655, 7623, 945, 21; 1, 323, 9570, 72254, 188188, 188188, 72254, 9570, 323, 1; 1, 525, 24675, 302359, 1345605, 2499861, 2036125, 715725, 99414, 4410, 36; 1, 808, 56896, 1055320, 7329975, 22338816, 32152848, 22338816, 7329975, 1055320, 56896, 808, 1; MAPLE A001263 := proc(n, k) if n <=0 or k <=0 then 0 ; elif k > n then 0 ; else binomial(n-1, k-1)*binomial(n, k-1)/k ; end if; end proc: A178657 := proc(n, k) (1-x)^(2*n-1)*add(A001263(n+2*l, 2*l+1)*x^l, l=0..20) ; expand(%) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Aug 30 2011 MATHEMATICA p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[2*k + n, 2*k] Binomial[ 2*k + n, 1 + 2*k]/(2*k + n))*x^k, {k, 0, Infinity}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 2, 10}]; Flatten[%] CROSSREFS Cf. A001263, A034839. Sequence in context: A144006 A193966 A113378 * A257490 A156289 A095922 Adjacent sequences:  A178654 A178655 A178656 * A178658 A178659 A178660 KEYWORD nonn,tabf AUTHOR Roger L. Bagula, Jun 01 2010 STATUS approved

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Last modified December 2 02:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)