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

1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923, 1512191781, 15914734443, 168802010001, 1802247516891, 19350710547021, 208783189719531, 2262263134211073, 24604815145831011, 268499713118585781, 2938736789722114731, 32250788066104022961

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Ignoring initial term, equals the logarithmic derivative of A206177.

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

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

Examples

			L.g.f.: L(x) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 +...
Exponentiation equals the g.f. of A206177:
exp(L(x)) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012.

Crossrefs

Cf. A206177, A000172, A206180, A216483, A216636.

Related to diagonal of rational functions: A268545-A268555.

Programs

Mathematica

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 *)
Table[HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -2], {n, 0, 20}] (* Jean-François Alcover, Oct 25 2019 *)

PARI

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

Sage

A206178 = lambda n: hypergeometric([-n,-n,-n], [1,1], -2)
[Integer(A206178(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 23 2014

Formula

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.

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

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

G.f.: hypergeom([1/3, 2/3],[1],54*x^2/(1-3*x)^3)/(1-3*x). - Mark van Hoeij, May 02 2013

a(n) = hypergeom([-n,-n,-n],[1,1], -2). - Peter Luschny, Sep 23 2014

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

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014