A089665 - OEIS

%I #17 May 27 2022 08:10:16

%S 0,4,73,788,6630,48120,316526,1940568,11284380,62968560,339954670,

%T 1786320184,9176663028,46248446608,229285525420,1120646918000,

%U 5409322603896,25824570392544,122086747617198,572130452101240,2660063893120900,12279619924999504,56318986959592676

%N a(n) = S2(n,2), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

%H G. C. Greubel, <a href="/A089665/b089665.txt">Table of n, a(n) for n = 0..1000</a>

%H Jun Wang and Zhizheng Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00206-1">On extensions of Calkin's binomial identities</a>, Discrete Math., 274 (2004), 331-342.

%F a(n) = (1/24)*n*( (n+1)*(7*n+5)*4^n - 2*(n-1)*(3*n^2 - 2*n + 1)*binomial(2*n, n)/(2*n-1) ). (See Wang and Zhang, p. 338.)

%F From _G. C. Greubel_, May 25 2022: (Start)

%F a(n) = (n/6)*( (n+1)*(7*n+5)*4^(n-1) - (n-1)*(3*n^2 - 2*n + 1)*Catalan(n-1) ).

%F G.f.: x*(4*(1+3*x) - x*(3 + 2*x + 4*x^2)*sqrt(1-4*x))/(1-4*x)^4.

%F E.g.f.: x*(4 + 22*x + 56*x^2/3)*exp(4*x) + (x^2/6)*exp(2*x)*( -(9 + 62*x + 145*x^2 + 84*x^3)*f(x, 0) + (36 + 99*x - 32*x^2 - 84 x^3)*f(x, 1) + (45 + 270*x + 284*x^2 + 48*x^3)*f(x, 2) + x*(109 + 224*x + 78*x^2)*f(x, 3) + x^2*(53 + 36*x)*f(x, 4) + 6*x^3*f(x, 5) ), where f(x, n) = BesselI(n, 2*x). (End)

%p S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);

%p seq(S2(n, 2), n = 0..40);

%t Table[(1/24)*(n*(n+1)*(7*n+5)*4^n -4*(n-1)*(3*n^2-2*n+1)*Binomial[2*n-2, n-1]), {n,0,40}] (* _G. C. Greubel_, May 25 2022 *)

%o (SageMath) [(n/6)*((n+1)*(7*n+5)*4^(n-1) -(n-1)*(3*n^2-2*n+1)*catalan_number(n-1)) for n in (0..40)] # _G. C. Greubel_, May 25 2022

%Y Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), this sequence (t=2), A089666 (t=3), A089667 (t=4), A089668 (t=5).

%Y Cf. A000108, A089658, A089669.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jan 04 2004