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A160562
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Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).
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6
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1, 1, 1, 1, 10, 1, 1, 91, 35, 1, 1, 820, 966, 84, 1, 1, 7381, 24970, 5082, 165, 1, 1, 66430, 631631, 273988, 18447, 286, 1, 1, 597871, 15857205, 14057043, 1768195, 53053, 455, 1, 1, 5380840, 397027996, 704652312, 157280838, 8187608, 129948, 680, 1
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OFFSET
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0,5
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COMMENTS
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This is table 4 on page 12 of Gelineau and Zeng, read downwards by columns.
Apparently the table can also be obtained by deleting each second row and column of A136630.
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LINKS
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FORMULA
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T(n,k) = (1/((2*k)!*4^k)) * Sum_{m=0..k} (-1)^(k-m)*A039599(k,m)*(2*m+1)^(2*n). - Werner Schulte, Nov 01 2015
T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). Note that sin(x)^(2*k+1) = (Sum_{i=0..k} (-1)^i*binomial(2*k+1,k-i)*sin((2*i+1)*x))/(2^(2*k)). - Jianing Song, Oct 29 2023
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EXAMPLE
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Triangle starts:
1;
1, 1;
1, 10, 1;
1, 91, 35, 1;
1, 820, 966, 84, 1;
...
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MAPLE
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A160562 := proc(n, k) npr := 2*n+1 ; kpr := 2*k+1 ; sinh(t*sinh(x)) ; npr!*coeftayl(%, x=0, npr) ; coeftayl(%, t=0, kpr) ; end: seq(seq(A160562(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Sep 09 2009
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MATHEMATICA
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T[n_, k_] := Sum[(-1)^(k - m)*(2m + 1)^(2n + 1)*Binomial[2k, k + m]/(k + m + 1), {m, 0, k}]/(4^k*(2k)!);
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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