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Search: id:A100247
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| A100247 |
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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]). |
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+0
3
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| 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; listen)
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OFFSET
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0,6
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COMMENT
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Row sums form A100248. Antidiagonal sums form A100249.
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FORMULA
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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.
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EXAMPLE
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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.
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PROGRAM
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(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)))} (PARI) {T(n, k)=polcoeff(((1-sqrt(1-4*z))/(2*z))^(n-k\2), k, z)}
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CROSSREFS
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Cf. A033184, A100248, A100249.
Sequence in context: A037010 A114695 A134084 this_sequence A011342 A084046 A016586
Adjacent sequences: A100244 A100245 A100246 this_sequence A100248 A100249 A100250
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 09 2004
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