%I #50 Sep 08 2022 08:45:10

%S 1,25,2641,392641,67982041,12838867105,2564949195985,533008982952625,

%T 114035552691160585,24950692835328410305,5557138347370070346601,

%U 1255741805437716400557625,287180884347761929741524361,66343186345544102086872515761

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

%C Diagonal of the rational function 1/(1-(x + y + z + w + x*y*z*w)). - _Gheorghe Coserea_, Jul 15 2016

%H Alois P. Heinz, <a href="/A082488/b082488.txt">Table of n, a(n) for n = 0..400</a>

%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>

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

%F 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

%F 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

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

%F 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

%F 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

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

%e where

%e 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]

%e Equivalently,

%e 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) +...

%p with(combinat):

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

%p seq(a(n), n=0..15); # _Alois P. Heinz_, Sep 23 2013

%t 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 *)

%o From _Paul D. Hanna_, Sep 22 2013: (Start)

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

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

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

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

%o (End)

%o (Magma) [&+[Binomial(n, k)*Binomial(n+k, k)*Binomial(n+2*k, k)*Binomial(n+3*k, k): k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Oct 16 2018

%Y Cf. A081798.

%Y Column k = 4 of A229142.

%Y Cf. A008977, A001850, A082489, A229049, A229050.

%Y Related to diagonal of rational functions: A268545-A268555.

%K easy,nonn

%O 0,2

%A _Emanuele Munarini_, Apr 28 2003