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A116388 Expansion of 1/((1+x*(1-M(x)))*sqrt(1-2*x-3*x^2)), M(x) the g.f. of A001006. 2
1, 1, 4, 10, 29, 82, 236, 681, 1975, 5745, 16757, 48982, 143442, 420721, 1235663, 3633453, 10695292, 31511524, 92919758, 274203662, 809719718, 2392579638, 7073684393, 20924387460, 61925598216, 183350728661, 543095661673 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

FORMULA

G.f.: 2*x/(sqrt(1-2*x-3*x^2)*(sqrt(1-2*x-3*x^2) -1 +2*x +3*x^2)).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j-k)*C(j,n-k-j).

Conjecture: n*a(n) + 7*(-n+1)*a(n-1) + 2*(4*n-9)*a(n-2) + (25*n-58)*a(n-3) + (-18*n+65)*a(n-4) + (-52*n+199)*a(n-5) + (-31*n+135)*a(n-6) + 6*(-n+5)*a(n-7) = 0. - R. J. Mathar, Jun 22 2016

MATHEMATICA

Table[Sum[Sum[Binomial[n-k, j-k]*Binomial[j, n-k-j], {j, 0, n-k}], {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, May 23 2019 *)

PROG

(PARI) {a(n) = sum(k=0, n\2, sum(j=0, n-k, binomial(n-k, j-k)*binomial(j, n-k-j)))}; \\ G. C. Greubel, May 23 2019

(MAGMA) [(&+[ (&+[Binomial(n-k, j-k)*Binomial(j, n-k-j): j in [0..n-k]]) : k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 23 2019

(Sage) [sum( sum(binomial(n-k, j-k)*binomial(j, n-k-j) for j in (0..n)) for k in (0..floor(n/2))) 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-k, j-k)*Binomial(j, n-k-j) ))) # G. C. Greubel, May 23 2019

CROSSREFS

Sequence in context: A052946 A026152 A025179 * A221420 A212262 A233347

Adjacent sequences:  A116385 A116386 A116387 * A116389 A116390 A116391

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 12 2006

STATUS

approved

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Last modified November 12 07:03 EST 2019. Contains 329052 sequences. (Running on oeis4.)