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 A131198 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,1,0,1,0,1,0,...] DELTA [0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938. 7
 1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Mirror image of triangle A090181, another version of triangle of Narayana (A001263). Equals A133336*A130595 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6 Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5 Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018. Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations , J. Int. Seq. 14 (2011) # 11.3.8 Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. FORMULA Sum_{k=0..n} T(n,k)*x^k = A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n), for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Oct 23 2007 Sum_{k=0..floor(n/2)} T(n-k,k) = A004148(n). - Philippe Deléham, Nov 06 2007 T(2*n,n) = A125558(n). - Philippe Deléham, Nov 16 2011 T(n, k) = [x^k] hypergeom([1 - n, -n], [2], x). - Peter Luschny, Apr 26 2022 EXAMPLE Triangle begins:   1;   1,  0;   1,  1,   0;   1,  3,   1,   0;   1,  6,   6,   1,   0;   1, 10,  20,  10,   1,   0;   1, 15,  50,  50,  15,   1,  0;   1, 21, 105, 175, 105,  21,  1, 0;   1, 28, 196, 490, 490, 196, 28, 1, 0; ... MAPLE T := (n, k) -> `if`(n=0, 0^n, binomial(n, k)^2*(n-k)/(n*(k+1))); seq(print(seq(T(n, k), k=0..n)), n=0..5); # Peter Luschny, Jun 08 2014 R := n -> simplify(hypergeom([1 - n, -n], [2], x)): Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n): seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022 MATHEMATICA Table[If[n == 0, 1, (n-k)*Binomial[n, k]^2/(n*(k+1))], {n, 0, 10}, {k, 0, n}] //Flatten (* G. C. Greubel, Feb 06 2018 *) PROG (PARI) for(n=0, 10, for(k=0, n, print1(if(n==0, 1, (n-k)*binomial(n, k)^2/(n* (k+1))), ", "))) \\ G. C. Greubel, Feb 06 2018 (MAGMA) [[n le 0 select 1 else (n-k)*Binomial(n, k)^2/(n*(k+1)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018 CROSSREFS Cf. A000217, A002415, A006542, A006857. Sequence in context: A059045 A348210 A122935 * A090181 A256551 A144417 Adjacent sequences:  A131195 A131196 A131197 * A131199 A131200 A131201 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Oct 20 2007 STATUS approved

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Last modified July 6 11:37 EDT 2022. Contains 355110 sequences. (Running on oeis4.)