login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k. 10
1, 3, 21, 171, 1521, 14283, 138909, 1385163, 14072193, 145039923, 1512191781, 15914734443, 168802010001, 1802247516891, 19350710547021, 208783189719531, 2262263134211073, 24604815145831011, 268499713118585781, 2938736789722114731, 32250788066104022961 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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.

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))^2/(2*2^(1/3)*sqrt(3)*Pi) * (3*2^(2/3)+3*2^(1/3)+3)^n/n. - Vaclav Kotesovec, Sep 19 2012

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

EXAMPLE

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 +...

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 *)

PROG

(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

CROSSREFS

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

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A132805 A189475 A331328 * A233861 A206397 A247480

Adjacent sequences:  A206175 A206176 A206177 * A206179 A206180 A206181

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2012

EXTENSIONS

Minor edits by Vaclav Kotesovec, Mar 31 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)