%I #48 Jul 11 2020 12:36:44

%S 1,4,34,352,3946,46744,573616,7217536,92527738,1203467464,15834369244,

%T 210304283776,2815055712496,37930536447808,513972867056704,

%U 6998587355233792,95704396144575898,1313665229153722408,18091969874675059204,249908773119244105792

%N a(n) = Sum_{k=0..n} binomial(n,k)^3 * 3^k.

%C Ignoring initial term, equals the logarithmic derivative of A206179.

%C Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.

%C Diagonal of rational function 1/(1 + y + z + x*y + y*z + 3*x*z + 4*x*y*z). - _Gheorghe Coserea_, Jul 01 2018

%C Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 3*x*y*z). - _Seiichi Manyama_, Jul 11 2020

%H Vincenzo Librandi, <a href="/A206180/b206180.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="http://www.kotesovec.cz/math_articles/kotesovec_binomial_asymptotics.pdf">Asymptotic of a sums of powers of binomial coefficients * x^k</a>, 2012.

%F Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 4*(9*n^3+57*n^2+116*n+74)*a(n+2) - (99*n^3+528*n^2+929*n+540)*a(n+1) - 64*(3*n+7)*(n+1)^2*a(n) = 0. - _Vaclav Kotesovec_, Sep 11 2012

%F a(n) ~ (1+3^(1/3))^2/(2*3^(5/6)*Pi) * (3*3^(2/3)+3*3^(1/3)+4)^n/n. - _Vaclav Kotesovec_, Sep 19 2012

%F G.f.: hypergeom([1/3, 2/3],[1],81*x^2/(1-4*x)^3)/(1-4*x). - _Mark van Hoeij_, May 02 2013

%F a(n) = hypergeometric([-n,-n,-n],[1,1], -3). - _Peter Luschny_, Sep 23 2014

%F G.f. y=A(x) satisfies: 0 = x*(2*x + 1)*(64*x^3 + 33*x^2 + 12*x - 1)*y'' + (384*x^4 + 388*x^3 + 123*x^2 + 24*x - 1)*y' + (128*x^3 + 132*x^2 + 24*x + 4)*y. - _Gheorghe Coserea_, Jul 01 2018

%e L.g.f.: L(x) = 4*x + 34*x^2/2 + 352*x^3/3 + 3946*x^4/4 + 46744*x^5/5 +...

%e Exponentiation equals the g.f. of A206179:

%e exp(L(x)) = 1 + 4*x + 25*x^2 + 196*x^3 + 1747*x^4 + 16996*x^5 + 175936*x^6 +...

%t Flatten[{1,RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-4*(9*n^3+57*n^2+116*n+74)*a[n+2]-(99*n^3+528*n^2+929*n+540)*a[n+1]-64*(3*n+7)*(n+1)^2*a[n]==0,a[1]==4,a[2]==34,a[3]==352},a,{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 11 2012 *)

%t Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -3] , {n, 0, 20}] (* _Jean-François Alcover_, Oct 25 2019 *)

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

%o for(n=0,41,print1(a(n),", "))

%o (Sage)

%o A206180 = lambda n: hypergeometric([-n,-n,-n],[1,1], -3)

%o [Integer(A206180(n).n(100)) for n in (0..19)] # _Peter Luschny_, Sep 23 2014

%Y Cf. A206179, A000172, A206178, A216483, A216636.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 04 2012

%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014