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A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k.

%I A206178 #61 Apr 24 2025 03:43:37
%S A206178 1,3,21,171,1521,14283,138909,1385163,14072193,145039923,1512191781,
%T A206178 15914734443,168802010001,1802247516891,19350710547021,
%U A206178 208783189719531,2262263134211073,24604815145831011,268499713118585781,2938736789722114731,32250788066104022961
%N A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k.
%C A206178 Ignoring initial term, equals the logarithmic derivative of A206177.
%C A206178 Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.
%C A206178 Diagonal of rational functions 1/(1 - x*y + y*z + 2*x*z - 3*x*y*z), 1/(1 + y + z + x*y + y*z + 2*x*z + 3*x*y*z), 1/(1 - x + 2*z + x*y - y*z - 2*x*z + 3*x*y*z), 1/(1 - x - y - z + x*y + y*z + x*z - 3*x*y*z), 1/(1 - x + y + 2*z - x*y + 2*y*z - 2*x*z - 3*x*y*z). - _Gheorghe Coserea_, Jul 03 2018
%H A206178 Vincenzo Librandi, <a href="/A206178/b206178.txt">Table of n, a(n) for n = 0..200</a>
%H A206178 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 A206178 a(2*3^n) == 3 (mod 9) for n>=0; a(n) == 0 (mod 9) if n/2 > 1 is not a power of 3.
%F A206178 Recurrence: (n+3)^2*(3*n+4)*a(n+3) - 3*(9*n^3+57*n^2+116*n+74)*a(n+2) - 3*(27*n^3+144*n^2+252*n+145)*a(n+1) - 27*(3*n+7)*(n+1)^2*a(n) = 0. - _Vaclav Kotesovec_, Sep 11 2012
%F A206178 a(n) ~ (1 + 2^(1/3))^(3*n + 2) / (2^(4/3)*sqrt(3)*Pi*n). - _Vaclav Kotesovec_, Sep 19 2012, simplified Apr 24 2025
%F A206178 G.f.: hypergeom([1/3, 2/3],[1],54*x^2/(1-3*x)^3)/(1-3*x). - _Mark van Hoeij_, May 02 2013
%F A206178 a(n) = hypergeom([-n,-n,-n],[1,1], -2). - _Peter Luschny_, Sep 23 2014
%F A206178 G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 27*x^2 + 9*x - 1)*y'' + (243*x^4 + 378*x^3 + 189*x^2 + 36*x - 2)*y' + 3*(x + 1)*(27*x^2 + 12*x + 2)*y. - _Gheorghe Coserea_, Jul 01 2018
%e A206178 L.g.f.: L(x) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 +...
%e A206178 Exponentiation equals the g.f. of A206177:
%e A206178 exp(L(x)) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +...
%t A206178 Flatten[{1,RecurrenceTable[{(n+3)^2*(3*n+4)*a[n+3]-3*(9*n^3+57*n^2+116*n+74)*a[n+2]-3*(27*n^3+144*n^2+252*n+145)*a[n+1]-27*(3*n+7)*(n+1)^2*a[n]==0, a[1]==3, a[2]==21, a[3]==171},a,{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 11 2012 *)
%t A206178 Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -2], {n, 0, 20}] (* _Jean-François Alcover_, Oct 25 2019 *)
%o A206178 (PARI) {a(n)=sum(k=0,n,binomial(n,k)^3*2^k)}
%o A206178 (Sage)
%o A206178 A206178 = lambda n: hypergeometric([-n,-n,-n], [1,1], -2)
%o A206178 [Integer(A206178(n).n(100)) for n in (0..20)] # _Peter Luschny_, Sep 23 2014
%Y A206178 Cf. A206177, A000172, A206180, A216483, A216636.
%Y A206178 Related to diagonal of rational functions: A268545-A268555.
%K A206178 nonn
%O A206178 0,2
%A A206178 _Paul D. Hanna_, Feb 04 2012
%E A206178 Minor edits by _Vaclav Kotesovec_, Mar 31 2014