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 A100247 Slanted Catalan convolution table, read by rows of 2*n+1 terms in row n, where T(n,k) = C(n+2*k-[k/2],k)*(n-[k/2])/(n+2*k-[k/2]). 2
 1, 1, 1, 0, 1, 2, 2, 5, 0, 1, 3, 5, 14, 14, 42, 0, 1, 4, 9, 28, 42, 132, 132, 429, 0, 1, 5, 14, 48, 90, 297, 429, 1430, 1430, 4862, 0, 1, 6, 20, 75, 165, 572, 1001, 3432, 4862, 16796, 16796, 58786, 0, 1, 7, 27, 110, 275, 1001, 2002, 7072, 11934, 41990, 58786, 208012 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row sums form A100248. Antidiagonal sums form A100249. LINKS FORMULA T(n, k) = A033184(n-[k/2], k) for n>0 (with A033184 formatted as a square array). G.f. A(x, y) satisfies: A(x^2, y) = ( (1+x)/(2*y - x*(1 - sqrt(1 - 4*x*y))) - (1-x)/(2*y + x*(1 - sqrt(1 + 4*x*y))) )*y/x. EXAMPLE Rows begin: [1], [1,1,0], [1,2,2,5,0], [1,3,5,14,14,42,0], [1,4,9,28,42,132,132,429,0], [1,5,14,48,90,297,429,1430,1430,4862,0], [1,6,20,75,165,572,1001,3432,4862,16796,16796,58786,0],... and is derived from the square array of Catalan convolutions (A033184) by shifting each column k down by [k/2] rows. PROG (PARI) {T(n, k) = if(n==k&k==0, 1, binomial(n+2*k-(k\2), k)*(n-(k\2))/(n+2*k-(k\2)))} for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print("")) (PARI) {T(n, k) = polcoeff(((1-sqrt(1-4*z +z*O(z^(2*n))))/(2*z))^(n-k\2), k, z)} for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A033184, A100248, A100249. Sequence in context: A012858 A292383 A332896 * A194123 A011342 A084046 Adjacent sequences: A100244 A100245 A100246 * A100248 A100249 A100250 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Nov 09 2004 STATUS approved

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Last modified December 9 11:21 EST 2022. Contains 358700 sequences. (Running on oeis4.)