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a(n) = Sum_{k=0..n} binomial(n+k,k)^4.
7

%I #42 Jul 15 2022 08:08:21

%S 1,17,1378,170257,25561876,4294835666,776487013506,147812510671121,

%T 29234435383857304,5955068493838815892,1241820686691538181636,

%U 263946916625793118532050,56996643356459050103185444,12473214064899644269110156626,2760963661677614009262282769378

%N a(n) = Sum_{k=0..n} binomial(n+k,k)^4.

%H Vincenzo Librandi, <a href="/A219562/b219562.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012

%F a(n) ~ 2^(8*n+4)/(15*Pi^2*n^2).

%F Recurrence: 4*(n-1)*(4*n-1)*(4*n+1)*(279825*n^6 - 2240985*n^5 + 7416081*n^4 - 12962383*n^3 + 12597634*n^2 - 6438500*n + 1347304)*n^4*a(n) = 2*(n-1)*(2290647450*n^12 - 22926837585*n^11 + 100717526436*n^10 - 254986993727*n^9 + 410380920831*n^8 - 435959897978*n^7 + 305660392723*n^6 - 134977315842*n^5 + 31413259700*n^4 + 2833672*n^3 - 2076143616*n^2 + 500898816*n - 39813120)*a(n-1) + (859902225*n^13 - 10755967005*n^12 + 60090860763*n^11 - 197381561581*n^10 + 422055067481*n^9 - 613861172995*n^8 + 615013106513*n^7 - 418396400175*n^6 + 182810864162*n^5 - 42759392772*n^4 + 146171272*n^3 + 2813432832*n^2 - 691172352*n + 55738368)*a(n-2) - 16*(n-2)^3*(2*n-3)^4*(279825*n^6 - 562035*n^5 + 408531*n^4 - 111409*n^3 - 5504*n^2 + 7968*n - 1024)*a(n-3).

%F G.f. as an expression in terms of 2F1 and 4F3 functions is given in the Maple program below. - _Mark van Hoeij_, Apr 02 2013

%p q := x-4+I*((x+4)*(16-x))^(1/2);

%p f := x*(q/8)^4;

%p s := ((q-2)/(8*I-6))^(1/4);

%p y1 := hypergeom([1/8, 1/8], [3/4], f) * s / x^(1/8);

%p r := 2/(x*((x+4)*(16-x))^(1/2)*y1^2);

%p h := hypergeom([1/2, 1/2, 1/2, 1/2],[1, 1, 1],256*x);

%p u := (15*(223*x+72)*x^2*diff(h,x,x,x)+(14579*x+3226)*x*diff(h,x,x)

%p +(9969*x+1002)*diff(h,x)+320*h)/(16*(16-x)*(x+4)*x^2);

%p ogf := y1^2*Int(r*(1+Int(r*Int(u/(r*y1)^2,x),x)),x) ;

%p # Check o.g.f. by computing a series expansion:

%p SER := proc(a,x) series(a,x,20) end:

%p INT := proc(a,x) int(SER(a,x),x) end:

%p SER(eval(ogf, Int = INT),x); # _Mark van Hoeij_, Apr 02 2013

%t Table[Sum[Binomial[n+k,k]^4, {k,0,n}], {n,0,20}]

%o (PARI) a(n) = sum(k=0, n, binomial(n+k,k)^4); \\ _Michel Marcus_, Jul 15 2022

%Y Cf. A001700, A112029, A112028, A219563, A219564.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 23 2012