A089667 a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

0, 4, 265, 5984, 85722, 944904, 8771462, 72095520, 541127988, 3785356752, 25032083230, 158102986624, 961123994220, 5656943319664, 32386277835772, 181019819948864, 990793669704552, 5323620638111136, 28137973407708174, 146552649537716992

Offset

0,2

Links

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.

Formula

a(n) = (1/480)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^n - 4*n*(n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*binomial(2*n, n)/((2*n-1)*(2*n-3)) ). (See Wang and Zhang, p. 338.)

From G. C. Greubel, May 25 2022: (Start)

a(n) = (1/30)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*Catalan(n-2) ).

G.f.: x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*sqrt(1-4*x) )/(1-4*x)^6. [Typo corrected by Georg Fischer, Nov 09 2022] (End)

Mathematica

Table[(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) -(n-1)*(15*n^5-99*n^3 + 116*n^2-34*n+6)*CatalanNumber[n-2]), {n, 0, 40}] (* G. C. Greubel, May 25 2022 *)
CoefficientList[Series[x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*Sqrt[1-4*x] )/(1-4*x)^6, {x, 0, 35}], x] (* Georg Fischer, Nov 09 2022 *)

Prog

(SageMath) [(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3+116*n^2-34*n+6)*catalan_number(n-2) ) for n in (0..40)] # G. C. Greubel, May 25 2022

Crossrefs

Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), A089666 (t=3), this sequence (t=4), A089668 (t=5).

Cf. A000108, A089658, A089669.

Sequence in context: A061788 A203839 A052136 * A119008 A357559 A108134

Adjacent sequences: A089664 A089665 A089666 * A089668 A089669 A089670

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 04 2004

Status

approved