A082488 - OEIS

A082488

a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k).

1, 25, 2641, 392641, 67982041, 12838867105, 2564949195985, 533008982952625, 114035552691160585, 24950692835328410305, 5557138347370070346601, 1255741805437716400557625, 287180884347761929741524361, 66343186345544102086872515761

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Diagonal of the rational function 1/(1-(x + y + z + w + x*y*z*w)). - Gheorghe Coserea, Jul 15 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

FORMULA

G.f.: Sum_{n>=0} (4*n)!/n!^4 * x^n / (1-x)^(4*n+1). - Paul D. Hanna, Sep 22 2013

Recurrence: n^3*(2*n-3)*(4*n-9)*(4*n-5)*a(n) = (4*n-9)*(4*n-3)*(520*n^4 - 1820*n^3 + 2109*n^2 - 905*n + 121)*a(n-1) - (192*n^6 - 1536*n^5 + 4748*n^4 - 7050*n^3 + 5065*n^2 - 1563*n + 171)*a(n-2) + (4*n-1)*(32*n^5 - 296*n^4 + 1040*n^3 - 1689*n^2 + 1209*n - 279)*a(n-3) - (n-3)^3*(2*n-1)*(4*n-5)*(4*n-1)*a(n-4). - Vaclav Kotesovec, Sep 23 2013

a(n) ~ c*d^n/(Pi^(3/2)*n^(3/2)), where d = 65 + 46*sqrt(2) + 2*sqrt(2*(1055 + 746*sqrt(2))) = 259.976980158726979... is the maximal positive root of the equation 1 - 4*d + 6*d^2 - 260*d^3 + d^4 = 0 and c = sqrt(8 + 5*sqrt(2) + sqrt(14*(11 + 8*sqrt(2))))/8 = 0.71529801573844067904424114047445568721... - Vaclav Kotesovec, Sep 23 2013, updated Jul 16 2016

G.f.: hypergeom([1/8, 3/8],[1],256*x/(1-x)^4)^2/(1-x). - Mark van Hoeij, Sep 23 2013

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 13*x^2 + 893*x^3 + 99125*x^4 + 13706093*x^5 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

0 = x^2*(3*x+1)^2*(1-260*x+6*x^2-4*x^3+x^4)*y''' + 3*x*(3*x+1)*(1-390*x-378*x^2+8*x^3-15*x^4+6*x^5)*y'' + (1-836*x+133*x^2+768*x^3-69*x^4-60*x^5+63*x^6)*y' + (-25+397*x-378*x^2-6*x^3+3*x^4+9*x^5)*y, where y is the g.f. - Gheorghe Coserea, Jul 15 2016

EXAMPLE

G.f.: A(x) = 1 + 25*x + 2641*x^2 + 392641*x^3 + 67982041*x^4 + 12838867105*x^5 +...

where

A(x) = 1/(1-x) + (4!/1!^4)*x/(1-x)^5 + (8!/2!^4)*x^2/(1-x)^9 + (12!/3!^4)*x^3/(1-x)^13 + (16!/4!^4)*x^4/(1-x)^17 + (20!/5!^4)*x^5/(1-x)^21 +... [Hanna]

Equivalently,

A(x) = 1/(1-x) + 24*x/(1-x)^5 + 2520*x^2/(1-x)^9 + 369600*x^3/(1-x)^13 + 63063000*x^4/(1-x)^17 + 11732745024*x^5/(1-x)^21 +...+ A008977(n)*x^n/(1-x)^(4*n+1) +...

MAPLE

with(combinat):

a:= n-> add(multinomial(n+3*k, n-k, k$4), k=0..n):

seq(a(n), n=0..15); # Alois P. Heinz, Sep 23 2013

MATHEMATICA

Table[Sum[Binomial[n, k]*Binomial[n+k, k]*Binomial[n+2*k, k]* Binomial[n+3*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 23 2013 *)

PROG

From Paul D. Hanna, Sep 22 2013: (Start)

(PARI) {a(n)=polcoeff(sum(m=0, n, (4*m)!/m!^4*x^m/(1-x+x*O(x^n))^(4*m+1)), n)}

for(n=0, 15, print1(a(n), ", "))

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k)*binomial(n+3*k, k))}

for(n=0, 15, print1(a(n), ", "))