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A116387 Expansion of 1/(sqrt(1-2*x-3*x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006. 3
1, 2, 7, 22, 72, 234, 763, 2486, 8099, 26372, 85833, 279226, 907946, 2951066, 9587981, 31140034, 101104048, 328162170, 1064856217, 3454513274, 11204337056, 36332719182, 117795920249, 381848062066, 1237615088203, 4010710218384 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A116383.

The substitution x-> x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform) yields the g.f. of A074331. - R. J. Mathar, Nov 10 2008

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j-k)*C(j,n-j).

Conjecture: n*(17*n-142)*a(n) + (17*n^2 + 95*n + 138)*a(n-1) + (-391*n^2 + 2488*n - 2908)*a(n-2) + (-17*n^2 - 603*n + 1892)*a(n-3) + 2*(697*n-2021)*(n-4)*a(n-4) + 60*(17*n-47)*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 15 2011

a(n) ~ (1+sqrt(5))^n * (5+sqrt(5)) / 10. - Vaclav Kotesovec, Feb 08 2014

MATHEMATICA

Table[Sum[Binomial[n, j-k]Binomial[j, n-j], {k, 0, n}, {j, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 08 2012 *)

PROG

(PARI) {a(n) = sum(k=0, n, sum(j=0, n, binomial(n, j-k)*binomial(j, n-j)))}; \\ G. C. Greubel, May 23 2019

(MAGMA) [(&+[ (&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2019

(Sage) [sum( sum(binomial(n, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, May 23 2019

(GAP) List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j) ))) # G. C. Greubel, May 23 2019

CROSSREFS

Sequence in context: A289592 A292230 A162770 * A294006 A322573 A294007

Adjacent sequences:  A116384 A116385 A116386 * A116388 A116389 A116390

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 12 2006

STATUS

approved

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Last modified November 21 22:16 EST 2019. Contains 329383 sequences. (Running on oeis4.)