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A155516
Triangle T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1), read by rows.
2
1, 1, 1, 1, 20, 1, 1, 105, 105, 1, 1, 336, 1764, 336, 1, 1, 825, 13860, 13860, 825, 1, 1, 1716, 70785, 226512, 70785, 1716, 1, 1, 3185, 273273, 2147145, 2147145, 273273, 3185, 1, 1, 5440, 866320, 14158144, 34763300, 14158144, 866320, 5440, 1, 1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1
OFFSET
0,5
FORMULA
T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*binomial(2*n, 2*k)*binomial(n, k).
From G. C. Greubel, May 29 2021: (Start)
T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*A155495(n, k).
T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1).
Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n,-n,-n-1/2,-n+1/2], [1/2,1,3/2], 1). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 20, 1;
1, 105, 105, 1;
1, 336, 1764, 336, 1;
1, 825, 13860, 13860, 825, 1;
1, 1716, 70785, 226512, 70785, 1716, 1;
1, 3185, 273273, 2147145, 2147145, 273273, 3185, 1;
1, 5440, 866320, 14158144, 34763300, 14158144, 866320, 5440, 1;
1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1;
MATHEMATICA
T[n_, k_]:= Binomial[2*n, 2*k]*Binomial[2*n+1, 2*k+1]/(2*n-2*k+1);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 29 2021 *)
PROG
(Magma) [Binomial(2*n, 2*k)*Binomial(2*n+1, 2*k+1)/(2*n-2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021
(Sage) flatten([[binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1) for k in (0..12)] for n in (0..12)]) # G. C. Greubel, May 29 2021
CROSSREFS
Cf. A155495.
Sequence in context: A040400 A139459 A154652 * A174674 A144443 A257608
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Jan 23 2009
EXTENSIONS
Edited by G. C. Greubel, May 29 2021
STATUS
approved