A089669 a(n) = S3(n,1), where S3(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^3.

0, 8, 155, 2286, 29296, 344140, 3807774, 40327280, 413058080, 4120742808, 40242188170, 386141947972, 3650905945872, 34087726136672, 314844824466704, 2880757518523200, 26141327872575616, 235490128979282224, 2107598857648209954, 18752794473550896332

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) = Sum_{k=0..n} k * (Sum_{j=0..k} binomial(n,j))^3. - G. C. Greubel, May 26 2022

From Vaclav Kotesovec, May 27 2022: (Start)

Recurrence: (n-3)*(n-2)*(n-1)*(81*n^5 - 1080*n^4 + 5769*n^3 - 15146*n^2 + 19080*n - 9088)*a(n) = (n-3)*(1863*n^7 - 29457*n^6 + 195435*n^5 - 696271*n^4 + 1410606*n^3 - 1569664*n^2 + 815328*n - 103936)*a(n-1) - 12*(1134*n^8 - 20709*n^7 + 162279*n^6 - 708529*n^5 + 1865571*n^4 - 2976218*n^3 + 2709336*n^2 - 1189824*n + 153600)*a(n-2) + 64*(405*n^8 - 7101*n^7 + 53712*n^6 - 228226*n^5 + 590469*n^4 - 934993*n^3 + 856278*n^2 - 392944*n + 65280)*a(n-3) + 256*(n-4)*(n-2)*(2*n - 7)*(81*n^5 - 675*n^4 + 2259*n^3 - 3509*n^2 + 2180*n - 384)*a(n-4).

a(n) ~ 3 * n^2 * 8^(n-1) * (1 - 1/sqrt(Pi*n) + (5/3 - 1/(2*Pi*sqrt(3)))/n). (End)

Mathematica

a[n_]:= a[n]= Sum[k*(Sum[Binomial[n, j], {j, 0, k}])^3, {k, 0, n}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, May 26 2022 *)

Prog

(SageMath)
@CachedFunction
def A089669(n): return sum(k*(sum(binomial(n, j) for j in (0..k)))^3 for k in (0..n))
[A089669(n) for n in (0..40)] # G. C. Greubel, May 26 2022

Crossrefs

Sequences of S3(n, t): A007403 (t=0), this sequence (t=1), A089670 (t=2), A089671 (t=3), A089672 (t=4).

Cf. A089658, A089664.

Sequence in context: A191908 A302959 A188408 * A288682 A268543 A345317

Adjacent sequences: A089666 A089667 A089668 * A089670 A089671 A089672

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 04 2004

Status

approved