login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A181069 L.g.f.: Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n. 2
1, 3, 28, 275, 3126, 37632, 475056, 6192531, 82754650, 1127504378, 15603575208, 218727171104, 3099183987004, 44315462038200, 638663235342528, 9267264584278419, 135279095477748642, 1985221072388231742 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..18.

FORMULA

a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * binomial(n,k).

Recurrence: (n-1)^2*n^3*(10*n^2 - 25*n + 16)*a(n) = 2*(n-1)^2*(60*n^5 - 240*n^4 + 341*n^3 - 225*n^2 + 90*n - 16)*a(n-1) + 4*(n-2)^2*n*(4*n - 7)*(4*n - 5)*(10*n^2 - 5*n + 1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014

a(n) ~ 2^(4*n-5/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Mar 06 2014

EXAMPLE

L.g.f.: L(x) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 +...

which equals the series:

L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x

+ (1 + 2^4*x + 3^4*x^2 + 4^4*x^3 + 5^4*x^4 + 6^4*x^5 +...)*x^2/2

+ (1 + 3^4*x + 6^4*x^2 + 10^4*x^3 + 15^4*x^4 + 21^4*x^5 +...)*x^3/3

+ (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 +...)*x^4/4

+ (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 + 70^4*x^4 + 126^4*x^5 +...)*x^5/5

+ (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 +...)*x^6/6

+ (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 +...)*x^7/7 +...

Exponentiation yields the g.f. of A181068:

exp(L(x)) = 1 + x + 2*x^2 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...

MATHEMATICA

Table[Sum[Binomial[n-1, k]^3 * Binomial[n, k], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)

PROG

(PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^4*n/(n-k))}

(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1, k)^4*x^k)*x^m/m)+x*O(x^n), n)}

for(n=1, 20, print1(a(n), ", "))

CROSSREFS

Cf. A181068, A181067 (variant).

Sequence in context: A076723 A198887 A026114 * A239297 A287884 A250890

Adjacent sequences:  A181066 A181067 A181068 * A181070 A181071 A181072

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 08 2010

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 July 19 08:48 EDT 2019. Contains 325155 sequences. (Running on oeis4.)