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A089666
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a(n) = S2(n,3), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
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5
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0, 4, 137, 2136, 23452, 209840, 1640346, 11648224, 76976048, 481048128, 2874897670, 16564931504, 92584313112, 504313834336, 2687067833492, 14045889333120, 72202366588096, 365713117287680, 1828223537042142, 9032706189007888, 44158716127799240, 213826835772518304
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OFFSET
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0,2
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
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FORMULA
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a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - (n-1)^2*n^2*(n+1)*binomial(2*n, n)/(8*(2*n-1)). (See Wang and Zhang, p. 338.)
From G. C. Greubel, May 25 2022: (Start)
a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - 9*binomial(n+1, 3)^2 * Catalan(n- 1)/(n+1).
G.f.: x*(4*(1 + 15*x + 12*x^2 + 8*x^3) - 3*x*(1 + 6*x - 6*x^2 + 4*x^3)*sqrt(1-4*x))/(1-4*x)^5. (End)
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MAPLE
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S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
seq(S2(n, 3), n = 0..40);
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MATHEMATICA
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Table[n*(15*n^3+30*n^2+21*n-2)*4^(n-3) -(n-1)^2*n^2*(n+1)*Binomial[2*n, n]/(8*(2*n -1)), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)
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PROG
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(SageMath) [n*(15*n^3+30*n^2+21*n-2)*4^(n-3) - 9*binomial(n+1, 3)^2 * catalan_number(n-1)/(n+1) for n in (0..40)] # G. C. Greubel, May 25 2022
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CROSSREFS
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Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), this sequence (t=3), A089667 (t=4), A089668 (t=5).
Cf. A000108, A089658, A089669.
Sequence in context: A229416 A155207 A201388 * A029850 A221675 A340838
Adjacent sequences: A089663 A089664 A089665 * A089667 A089668 A089669
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Jan 04 2004
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EXTENSIONS
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Name changed by G. C. Greubel, May 25 2022
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STATUS
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approved
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