A226705 - OEIS

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.

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

From Seiichi Manyama, Aug 05 2025: (Start)

a(n) = [x^n] 1/((1+2*x) * (1-x)^(5*n+1)).

a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(6*n+1,k).

a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+k,k). (End)

From Seiichi Manyama, Aug 14 2025: (Start)

a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).

G.f.: G(x)^2/((-2+3*G(x)) * (6-5*G(x))) where G(x) = 1+x*G(x)^6 is the g.f. of A002295. (End)

G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A004355. - Seiichi Manyama, Aug 15 2025

D-finite with recurrence: -15381809912925388800*(6*n + 5)*(3*n + 2)*(2*n + 1)*(3*n + 1)*(6*n + 7)*a(n) - 655360*(19578440952132544*n^5 - 154883488873737812*n^4 - 1031574315463211038*n^3 - 2134011905053822177*n^2 - 1928873900061230067*n - 656816150665058670)*a(n + 1) + 10240*(6828132066794577968*n^5 + 49700791681017041516*n^4 + 92058104249633732122*n^3 - 117524092240563958343*n^2 - 537720627594848899593*n - 446445215744894745480)*a(n + 2) - 1440*(64524467659456349312*n^5 + 973237165497923357348*n^4 + 5867297321045488101814*n^3 + 17608974094159346776201*n^2 + 26203769357271950650155*n + 15392191089791444953320)*a(n + 3) + 324*(25533355283369839271*n^5 + 445919661269133118280*n^4 + 3152798558577235857940*n^3 + 11307441866065938348040*n^2 + 20613142009527214022649*n + 15301831793079995023140)*a(n + 4) - 2187*(246201601429780811*n^5 + 6354770730801889490*n^4 + 66242679735270821965*n^3 + 348230486504489717650*n^2 + 922203740928724228104*n + 983241834744391750080)*a(n + 5) + 40770492147746235*(5*n + 28)*(5*n + 29)*(n + 6)*(5*n + 26)*(5*n + 27)*a(n + 6) = 0. - Robert Israel, Mar 16 2026

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).

MAPLE

f:= gfun:-rectoproc({-15381809912925388800*(6*n + 5)*(3*n + 2)*(2*n + 1)*(3*n + 1)*(6*n + 7)*a(n) - 655360*(19578440952132544*n^5 - 154883488873737812*n^4 - 1031574315463211038*n^3 - 2134011905053822177*n^2 - 1928873900061230067*n - 656816150665058670)*a(n + 1) + 10240*(6828132066794577968*n^5 + 49700791681017041516*n^4 + 92058104249633732122*n^3 - 117524092240563958343*n^2 - 537720627594848899593*n - 446445215744894745480)*a(n + 2) - 1440*(64524467659456349312*n^5 + 973237165497923357348*n^4 + 5867297321045488101814*n^3 + 17608974094159346776201*n^2 + 26203769357271950650155*n + 15392191089791444953320)*a(n + 3) + 324*(25533355283369839271*n^5 + 445919661269133118280*n^4 + 3152798558577235857940*n^3 + 11307441866065938348040*n^2 + 20613142009527214022649*n + 15301831793079995023140)*a(n + 4) - 2187*(246201601429780811*n^5 + 6354770730801889490*n^4 + 66242679735270821965*n^3 + 348230486504489717650*n^2 + 922203740928724228104*n + 983241834744391750080)*a(n + 5) + 40770492147746235*(5*n + 28)*(5*n + 29)*(n + 6)*(5*n + 26)*(5*n + 27)*a(n + 6), a(0) = 1, a(1) = 4, a(2) = 48, a(3) = 600, a(4) = 7856, a(5) = 105684 }, a(n), remember):

map(f, [$0..30]); # Robert Israel, Mar 16 2026

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), ", "))