A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

0, 1, 20, 234, 2144, 16750, 117432, 761460, 4654848, 27173718, 152867000, 834212236, 4438175040, 23108423884, 118111709744, 594059985000, 2946077521920, 14429322555750, 69892354873080, 335194270938780, 1593211647720000, 7511501237722020, 35153884344493200

Offset: 0

Author

Paul Boddington, Mar 05 2003

Keywords

• easy
• nonn

References

• H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).

Crossrefs

Cf. A000108 (Catalan numbers).

Cf. A000984, A002457, A037966, A037972, A329444, A329913.

Programs

Magma

[n le 1 select n else n^2*(n^3+n^2-3*n-1)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, Jun 23 2022

Mathematica

Total/@Table[r^4 Binomial[n,r]^2,{n,0,20},{r,n}] (* Harvey P. Dale, Dec 04 2017 *)
Table[n^2*(n^3+n^2-3*n-1)*CatalanNumber[n-2] -Boole[n==1], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)

PARI

vector(30, n, n--; sum(k=1, n, k^4*binomial(n,k)^2)) \\ Michel Marcus, Aug 19 2015

SageMath

[n^2*(n^3+n^2-3*n-1)*catalan_number(n-2) for n in (0..30)] # G. C. Greubel, Jun 23 2022

Formula

For n>1 a(n) = n^2*(n^3+n^2-3*n-1)*C(n-2). Here C(n-2) = binomial(2*n-4, n-2)/(n-1) is a Catalan number.

From G. C. Greubel, Jun 23 2022: (Start)

a(n) = (n^2*(n^3 + n^2 - 3*n -1)/(2*(2*n-3)))*binomial(2*n-2, n-1).

G.f.: x*(1 + 2*x + 32*x^3 - 128*x^4 + 144*x^5)/(1-4*x)^(9/2).

E.g.f.: x*exp(2*x)*( (1+2*x)*(1 +6*x +4*x^2)*BesselI(0, 2*x) + 2*x*(2 + 7*x + 4*x^2)*BesselI(1, 2*x) ). (End)

D-finite with recurrence (n-1)*(39*n-106)*a(n) +4*(-38*n^2+n+290)*a(n-1) +4*(100*n^2-784*n+1145)*a(n-2) -64*(13*n+4)*(2*n-9)*a(n-3)=0. - R. J. Mathar, Sep 13 2024

Extensions

Terms a(18) and beyond from Andrew Howroyd, Jan 16 2020