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 A226705 G.f.: 1 / (1 + 12*x*G(x)^4 - 16*x*G^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295. 4
 1, 4, 48, 600, 7856, 105684, 1447392, 20075416, 281086416, 3964453368, 56240518128, 801624722232, 11470976280960, 164691196943212, 2371222443727584, 34224696393237360, 495036708728067088, 7173892793100898728, 104135761805147016096, 1513892435551302963792 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n) = Sum_{k=0..n} C(2*k, n-k) * C(6*n-2*k, k). a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(5*n-2*k, k). a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(4*n-2*k, k). a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(3*n-2*k, k). a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(2*n-2*k, k). a(n) = Sum_{k=0..n} C(5*n+2*k, n-k) * C(n-2*k, k). a(n) = Sum_{k=0..n} C(6*n+2*k, n-k) * C(-2*k, k). Self-convolution of A226706. G.f.: 1 / (1 - 4*x*G(x)^4 - 16*x^2*G(x)^10) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295. a(n) ~ 2^(6*n-2)*3^(6*n+3/2)/(5^(5*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 16 2013 EXAMPLE G.f.: A(x) = 1 + 4*x + 48*x^2 + 600*x^3 + 7856*x^4 + 105684*x^5 +... A related series is G(x) = 1 + x*G(x)^6, where G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +... G(x)^4 = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +... G(x)^5 = 1 + 5*x + 40*x^2 + 385*x^3 + 4095*x^4 + 46376*x^5 +... such that A(x) = 1/(1 + 12*x*G(x)^4 - 16*x*G^5). MATHEMATICA Table[Sum[Binomial[3*n+2*k, n-k]*Binomial[3*n-2*k, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 16 2013 *) PROG (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^6+x*O(x^n)); polcoeff(1/(1+12*x*G^4-16*x*G^5), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^6+x*O(x^n)); polcoeff(1/(1-4*x*G^4-16*x^2*G^10), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(6*n-2*k, k))} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=sum(k=0, n, binomial(3*n +2*k, n-k)*binomial(3*n-2*k, k))} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=sum(k=0, n, binomial(6*n +2*k, n-k)*binomial(-2*k, k))} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A226706, A183160, A226733, A147855, A002295. Sequence in context: A196963 A220325 A265419 * A126967 A098402 A333481 Adjacent sequences:  A226702 A226703 A226704 * A226706 A226707 A226708 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 15 2013 STATUS approved

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Last modified June 19 18:24 EDT 2021. Contains 345144 sequences. (Running on oeis4.)