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!)
A114496 a(n) = Sum of binomial(n,k)*binomial(2n+k,k) over all k. 9
1, 4, 26, 190, 1462, 11584, 93536, 765314, 6323270, 52638760, 440815036, 3709445084, 31340292076, 265683004240, 2258793820988, 19251776923210, 164440378882630, 1407266585304760, 12063701803046300, 103571977632247076 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Modification of A001850 inspired by the Apéry numbers A005259.

From Paul Barry, Feb 17 2009: (Start)

Central coefficient of (1 + 4x + 5x^2 + 2x^3)^n. The coefficients are the 4th row of A029635.

The third row of A029635 corresponds to the central Delannoy numbers A001850. (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

FORMULA

a(n) = Sum_{k=0..n} (binomial(n,k)*binomial(2n+k,k)).

Recurrence: 20*n*(2*n - 1)*a(n) = (371*n^2 - 411*n + 120)*a(n-1) -2*(81*n^2 - 299*n + 278)*a(n-2) + 4*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ sqrt(1734 + 442*sqrt(17))*((71 + 17*sqrt(17))/16)^n/(68*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

From Peter Bala, Oct 05 2015: (Start)

a(n) = Sum_{i = 0..n} 2^(n-i)*binomial(2*n,i)*binomial(n,i).

4*n*(2*n - 1)*(17*n - 23)*a(n) = (1207*n^3 - 2840*n^2 + 1897*n - 360)*a(n-1) - 2*(n - 1)*(17*n - 6)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 4.

1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + ... is the o.g.f. for A003168. (End)

MATHEMATICA

Table[Sum[Binomial[n, k]*Binomial[2n+k, k], {k, 0, n}], {n, 0, 25}]

PROG

(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n, k)*binomial(n, k));

vector(50, n, a(n-1)) \\ Altug Alkan, Oct 05 2015

CROSSREFS

Cf. A114497, A114498, A003168, A156894.

Cf. A156886. - Paul Barry, Feb 17 2009

Sequence in context: A052763 A213101 A084211 * A127086 A198024 A278393

Adjacent sequences:  A114493 A114494 A114495 * A114497 A114498 A114499

KEYWORD

nonn,easy

AUTHOR

Eric Rowland, Dec 01 2005

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 29 19:26 EST 2020. Contains 338769 sequences. (Running on oeis4.)