login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 05:38 EST 2022. Contains 358431 sequences. (Running on oeis4.)