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 A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n. 1
 1, 5, 1, 45, 10, 1, 505, 115, 15, 1, 6345, 1460, 210, 20, 1, 85405, 19765, 2990, 330, 25, 1, 1204245, 279710, 43635, 5220, 475, 30, 1, 17558705, 4088615, 651165, 81955, 8275, 645, 35, 1, 262577745, 61254760, 9901860, 1290520, 139350, 12280, 840, 40, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50) FORMULA T(n, k) = Sum_{j = k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1). - Detlef Meya, Jan 15 2024 EXAMPLE Triangle starts: [0] 1 [1] 5, 1 [2] 45, 10, 1 [3] 505, 115, 15, 1 [4] 6345, 1460, 210, 20, 1 [5] 85405, 19765, 2990, 330, 25, 1 [6] 1204245, 279710, 43635, 5220, 475, 30, 1 MATHEMATICA T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten T[n_, k_] := Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}]; Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Jan 15 2024 *) PROG (PARI) T(n, k) = sum(j = k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1)) \\ Andrew Howroyd, Jan 15 2024 CROSSREFS T(n, 0) = A133305(n). Row sums are A297705, alternating row sums are A131765. Cf. A103209. Sequence in context: A134275 A264774 A114154 * A134273 A048897 A049029 Adjacent sequences: A297896 A297897 A297898 * A297900 A297901 A297902 KEYWORD nonn,tabl AUTHOR Peter Luschny, Jan 08 2018 STATUS approved

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Last modified June 18 23:32 EDT 2024. Contains 373488 sequences. (Running on oeis4.)