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 A147855 G.f.: 1 / (1 + 4*x*G(x)^2 - 7*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. 5
 1, 3, 22, 174, 1444, 12323, 107104, 942952, 8381596, 75053100, 676017962, 6118171326, 55591175956, 506805088026, 4633571685968, 42468065811884, 390071875757852, 3589637747968964, 33089300640166360, 305476314574338648, 2823932709938708824, 26137341654281261347 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k). a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k). a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k). a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k). a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k). G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - Vaclav Kotesovec, Jun 16 2013 EXAMPLE G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +... A related series is G(x) = 1 + x*G(x)^4, where G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +... G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +... G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +... such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3). MATHEMATICA Table[Sum[Binomial[2*n+k, n-k]*Binomial[2*n-k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 16 2013 *) PROG (PARI) {a(n)=sum(k=0, n, binomial(2*n+k, n-k)*binomial(2*n-k, k))} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-k, k))} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+4*x*G^2-7*x*G^3), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A226733, A226761, A226705, A183160, A002293. Sequence in context: A074576 A077244 A138899 * A278333 A132595 A065204 Adjacent sequences:  A147852 A147853 A147854 * A147856 A147857 A147858 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 16 2013 STATUS approved

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Last modified December 8 04:46 EST 2019. Contains 329853 sequences. (Running on oeis4.)