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 A187056 G.f.: A(x,y,z) = Sum_{n>=0} ((2n)!/n!^2)*[Sum_{k=0..2n} T(n,k)*z^k]*x^(2n)*y^n/(1-x-xy)^(4n+1) where A(x,y,x+xy) = Sum_{n>=0, k=0..n} C(n,k)^4*x^n*y^k at z = x+xy; this is the triangle of coefficients T(n,k), read by rows. 0
 1, 7, 4, 1, 131, 176, 96, 16, 1, 3067, 6588, 5895, 2416, 477, 36, 1, 79459, 235456, 298816, 197824, 73120, 14656, 1504, 64, 1, 2181257, 8252300, 13668975, 12563200, 6966400, 2373504, 490700, 58400, 3675, 100, 1, 62165039, 286326288, 587324232, 692965040, 516541455, 252283968, 81432456, 17138304, 2276145, 179440, 7632, 144, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA Row sums equal A000897(n) = (4n)!/((2n)!*n!^2). Column 0 equals A099601(n) = quotient of de Bruijn sums S(4,n)/S(2,n). EXAMPLE G.f.: A(x,y,z) = 1/(1-x-x*y) + 2*(7 + 4*z + z^2)*x^2*y/(1-x-x*y)^5 + 6*(131 + 176*z + 96*z^2 + 16*z^3 + z^4)*x^4*y^2/(1-x-x*y)^9 + 20*(3067 + 6588*z + 5895*z^2 + 2416*z^3 + 477*z^4 + 36*z^5 + z^6)*x^6*y^3/(1-x-x*y)^13 +... G.f. at z = x+xy yields: A(x,y,x+xy) = 1 + (1 + y)*x + (1 + 16*y + y^2)*x^2 + (1 + 81*y + 81*y^2 + y^3)*x^3 + (1 + 256*y + 1296*y^2 + 256*y^3 + y^4)*x^4 + (1 + 625*y + 10000*y^2 + 10000*y^3 + 625*y^4 + y^5)*x^5 +... which is a series involving binomial coefficients to the 4-th power. ... This triangle of coefficients T(n,k) of z^k, k=0..2n, begins: [1]; [7, 4, 1]; [131, 176, 96, 16, 1]; [3067, 6588, 5895, 2416, 477, 36, 1]; [79459, 235456, 298816, 197824, 73120, 14656, 1504, 64, 1]; [2181257, 8252300, 13668975, 12563200, 6966400, 2373504, 490700, 58400, 3675, 100, 1]; [62165039, 286326288, 587324232, 692965040, 516541455, 252283968, 81432456, 17138304, 2276145, 179440, 7632, 144, 1]; [1818812387, 9876304172, 24205612067, 34939683632, 32837525567, 21029302364, 9356637759, 2899564224, 619135629, 88879924, 8237341, 461776, 14161, 196, 1]; [54257991011, 339398092544, 968547444480, 1655445817088, 1881608595776, 1496188189440, 853911382016, 353544477440, 106191762336, 22927328512, 3492995968, 364541184, 24932320, 1044736, 24192, 256, 1]; ... CROSSREFS Cf. A000897, A099601. Sequence in context: A011407 A021907 A168422 * A103240 A155531 A216261 Adjacent sequences:  A187053 A187054 A187055 * A187057 A187058 A187059 KEYWORD nonn,tabf AUTHOR Paul D. Hanna, Mar 02 2011 STATUS approved

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Last modified May 22 08:19 EDT 2013. Contains 225512 sequences.