cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A204820 a(n) = -4*a(n-1)*A001505(n-2), with a(1)=8.

Original entry on oeis.org

8, -192, 161280, -638668800, 6974263296000, -162193467211776000, 6893871130369327104000, -483949753351926762700800000, 52208499391605859160162304000000, -8200911084433448356878294712320000000
Offset: 1

Views

Author

John M. Campbell, Jan 19 2012

Keywords

Comments

Sums of coefficients from (4n+1)th moments of binomial(m,k) * binomial(3*m,k); see Maple code below.

Examples

			The evaluation of sum(binomial(n, k)*binomial(3*n, k)*k^9, k=0..n) involves the polynomial 2187*n^11+6561*n^10-45927*n^9-28431*n^8+322947*n^7-257985*n^6-473445*n^5+726003*n^4-110482*n^3-189924*n^2+52624*n-4320, the sum of the coefficients of which is -192 = a(2).

Links

Crossrefs

Cf. A203778, A015219, A202948, A202946.

Programs

  • Maple
    with(PolynomialTools); polyn:=q->expand(simplify((1/(GAMMA(n-((2*floor((q+1)/4)-1))/(2))))*(1/sqrt(3))*GAMMA(n+1/3)*GAMMA(n+2/3)*(1/3)*(1/(27^(-n)))*GAMMA(n)*1/64^n*sum(binomial(n, k)*binomial(3*n, k)*k^q, k=0..n)*(1/(GAMMA(2*n-((2*floor(q/2)-1)/(2)))))*(2^((floor((1/2)*q+1/2)-1)+q)))); coefl:=h->CoefficientList(expand(polyn(h)), n); coe:=(d, b)->coefl(d)[b];seq(sum(coe((4*d+1),b),b=1..(4*d+1)+floor(((4*d+1)+1)/4)+floor((4*d+1)/2)),d=1..6);seq(-(1/8)*GAMMA(2*n-3/2)*GAMMA(n-1/2)*(-1)^n*64^n/Pi,n=1..6);

Formula

a(n)=-(1/8)*GAMMA(2*n-3/2)*GAMMA(n-1/2)*(-1)^n*64^n/Pi

A205795 Sums of coefficients of polynomials from 5n-th moments of X ~ Hypergeometric(4m, 5m, m).

Original entry on oeis.org

24, 2880, 43545600, 5230697472000, 2432902008176640000, 3102242008666197196800000, 8841761993739701954543616000000, 49205466506600690141269768273920000000, 485663859076129603777149565235783270400000000, 7911522544013240381082219675638737768808448000000000
Offset: 1

Views

Author

John M. Campbell, Feb 09 2012

Keywords

Comments

See Maple code below for formula for such polynomials.

Examples

			The evaluation of sum(binomial(n, k)*binomial(4*n, k)*k^5, k = 0 .. n) involves the polynomial  256*n^5-640*n^3+400*n^2+108*n-100, the sum of the coefficients of which is 24 = a(1).

Links

Crossrefs

Cf. A204820, A203778, A015219, A202948, A202946.

Programs

  • Maple
    with(PolynomialTools);polyn:=w->simplify(Pi^2*sum(binomial(n,k)*binomial(4*n,k)*k^w,k=0..n)*5^w/3125^n*csc((1/5)*Pi)*csc((2/5)*Pi)*GAMMA(4*n)/GAMMA(n-(floor((w+1)/5)*5-2)/5)/GAMMA(n-(floor(w/5)*5-1)/5)/GAMMA(n-(floor((w+2)/5)*5-3)/5)/GAMMA(n-(floor((w+3)/5)*5-4)/5));coefl:=d->CoefficientList(expand(polyn(d)),n);seq(sum(coefl(5*h)[m],m=1..nops(coefl(5*h))),h=1..5);seq(simplify(12*5^(5*n-5)*GAMMA(n-4/5)*GAMMA(n-3/5)*GAMMA(n-2/5)*GAMMA(n-1/5)*(cos((2/5)*Pi)+cos((1/5)*Pi))/Pi^2),n=1..5);

Formula

a(n) = 120*A151989(n-2)*a(n-1), with a(1)=24.
a(n) = 12*5^(5*n-5)*GAMMA(n-4/5)*GAMMA(n-3/5)*GAMMA(n-2/5)*GAMMA(n-1/5)*(cos((2/5)*Pi)+cos((1/5)*Pi))/Pi^2.
Showing 1-2 of 2 results.

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