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 A008550 Table T(n,k), n>=0 and k>=0, read by antidiagonals: the k-th column given by the k-th Narayana polynomial. 9
 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 11, 4, 1, 1, 1, 42, 45, 19, 5, 1, 1, 1, 132, 197, 100, 29, 6, 1, 1, 1, 429, 903, 562, 185, 41, 7, 1, 1, 1, 1430, 4279, 3304, 1257, 306, 55, 8, 1, 1, 1, 4862, 20793, 20071, 8925, 2426, 469, 71, 9, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Mirror image of A243631. - Philippe Deléham, Sep 26 2014 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..11475 H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, arXiv:math/0103149 [math.CO], 2001. H. Prodinger, On a functional difference equation of Runyon, Morrison, Carlitz, and Riordan, Séminaire Lotharingien de Combinatoire 46 (2001), Article B46a. FORMULA T(n, k) = Sum_{j>0} A001263(k, j)*n^(j-1); T(n, 0)=1. T(n, k) = Sum_{j, 0<=j<=k} A088617(k, j)*n^j*(1-n)^(k-j). The o.g.f. of row n is gf(n) = 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1). - Peter Luschny, Nov 17 2014 G.f. of row n: 1/(1 - x/(1 - n*x/(1 - x/(1 - n*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017 EXAMPLE Row n=0: 1, 1, 1, 1, 1, 1, 1, 1, ... see A000012. Row n=1: 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108. Row n=2: 1, 1, 3, 11, 45, 197, 903, 4279, ... see A001003. Row n=3: 1, 1, 4, 19, 100, 562, 3304, 20071, ... see A007564. Row n=4: 1, 1, 5, 29, 185, 1257, 8925, ... see A059231. Row n=5: 1, 1, 6, 41, 306, 2426, 20076, ... see A078009. Row n=6: 1, 1, 7, 55, 469, 4237, 39907, ... see A078018. Row n=7: 1, 1, 8, 71, 680, 6882, 72528, ... see A081178. Row n=8: 1, 1, 9, 89, 945, 10577, 123129, ... see A082147. Row n=9: 1, 1, 10, 109, 1270, 15562, 198100, ... see A082181. Row n=10: 1, 1, 11, 131, 161, 1661, 22101, ... see A082148. Row n=11: 1, 1, 12, 155, 2124, 30482, 453432, ... see A082173. ... - Philippe Deléham, Apr 03 2013 MAPLE gf := n -> 2/(sqrt((n-1)^2*x^2-2*(n+1)*x+1)+(n-1)*x+1): for n from 0 to 11 do PolynomialTools:-CoefficientList(convert( series(gf(n), x, 12), polynom), x) od; # Peter Luschny, Nov 17 2014 MATHEMATICA Unprotect[Power]; Power[0 | 0, 0 | 0] = 1; Protect[Power]; Table[Function[n, Sum[Apply[Binomial[#1 + #2, #1] Binomial[#1, #2]/(#2 + 1) &, {k, j}]*n^j*(1 - n)^(k - j), {j, 0, k}]][m - k + 1] /. k_ /; k <= 0 -> 1, {m, -1, 9}, {k, m + 1, 0, -1}] // Flatten (* Michael De Vlieger, Aug 10 2017 Note: this code renders 0^0 = 1. To restore normal Power functionality: Unprotect[Power]; ClearAll[Power]; Protect[Power] *) CROSSREFS Columns: A000012, A000012, A000027, A028387, A090197, A090198, A090199, A090200. Main diagonal is A242369. A diagonal is in A099169. Cf. A204057 (another version), A088617, A243631. Sequence in context: A155586 A069739 A066060 * A064094 A090182 A256384 Adjacent sequences:  A008547 A008548 A008549 * A008551 A008552 A008553 KEYWORD easy,nonn,tabl AUTHOR Philippe Deléham, Jan 23 2004 STATUS approved

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Last modified January 19 03:18 EST 2019. Contains 319282 sequences. (Running on oeis4.)