cp's OEIS Frontend

G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137952.

a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

Recurrence: 5*n*(5*n-3)*(5*n-2)*(5*n+1)*(5*n+4)*(2948400*n^11 - 80922240*n^10 + 991552680*n^9 - 7191167904*n^8 + 34388915791*n^7 - 113938412552*n^6 + 266574560812*n^5 - 439214051186*n^4 + 497527715029*n^3 - 367402366838*n^2 + 158427508008*n - 30063700800)*a(n) = -240*(5*n-1)*(3402000*n^13 - 102564900*n^12 + 1682146080*n^11 - 16176231033*n^10 + 95359496344*n^9 - 359981654612*n^8 + 893831335718*n^7 - 1468770570635*n^6 + 1566970769558*n^5 - 1019176919948*n^4 + 331927521052*n^3 + 34505928*n^2 - 32180612832*n + 6541274880)*a(n-1) + 180*(884520000*n^16 - 26930232000*n^15 + 372745486800*n^14 - 3118060887120*n^13 + 17644263763548*n^12 - 71507400823524*n^11 + 214013670957835*n^10 - 480132169105811*n^9 + 810380315383846*n^8 - 1022562903644722*n^7 + 947982058983979*n^6 - 624324084479227*n^5 + 273663045967416*n^4 - 68343334466444*n^3 + 4273926176256*n^2 + 2065304121408*n - 381518968320)*a(n-2) + 72*(5890903200*n^16 - 188191699920*n^15 + 2743292998800*n^14 - 24248455085592*n^13 + 145518104758338*n^12 - 628264374415281*n^11 + 2014705595228766*n^10 - 4876859081303636*n^9 + 8950855221646414*n^8 - 12378944029917433*n^7 + 12665670452628658*n^6 - 9249292270917382*n^5 + 4496305419163048*n^4 - 1229711760456116*n^3 + 68797455703176*n^2 + 53468550934560*n - 10544040864000)*a(n-3) + 72*(5731689600*n^16 - 191702972160*n^15 + 2927459413440*n^14 - 27105381081216*n^13 + 170350803352728*n^12 - 770345146059408*n^11 + 2589617705669352*n^10 - 6581794624393248*n^9 + 12710327685293639*n^8 - 18531898603387194*n^7 + 20012311600272546*n^6 - 15421584075698196*n^5 + 7904537517669183*n^4 - 2290793383663938*n^3 + 159318295564312*n^2 + 94065554487360*n - 19593691084800)*a(n-4) + 72*(2*n-9)*(3*n-11)*(3*n-7)*(6*n-25)*(6*n-23)*(2948400*n^11 - 48489840*n^10 + 344492280*n^9 - 1422208584*n^8 + 3817772239*n^7 - 6909787807*n^6 + 8311308487*n^5 - 6272196721*n^4 + 2621759746*n^3 - 403021048*n^2 - 67705152*n + 22579200)*a(n-5). - Vaclav Kotesovec, Mar 25 2014

a(n) ~ sqrt(3*s*(s-1)*(3*s-2)/(5*s-3)) / (2*sqrt(Pi)*n^(3/2)*r^n), where s = 1.7888356349988794022183... is the root of the equation 216*(s-1)^2 = s*(5*s-6)^4, and r = 1/(s*(5*s-6)) = 0.189873988477346598... - Vaclav Kotesovec, Mar 25 2014

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137956.

a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(2*s*(1-s)*(4-5*s) / ((56*s - 48)*Pi)) / (n^(3/2) * r^n), where r = 0.1569043698639381952962655091205241634381480571697... and s = 1.444765371242615455251538467189577278901629278244... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^2, 8 * r^2 * s^3 * (1 + r*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137968.

a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(2*s*(1-s)*(6-7*s) / ((132*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.297009871974239150024579315539982910111693413337... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^2, 12 * r^2 * s^5 * (1 + r*s^6) = 1. - Vaclav Kotesovec, Nov 22 2017

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137961.

a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(2*s*(1-s)*(5-6*s) / ((90*s - 80)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.354660923650925199331121807468321286698258863972... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^2, 10 * r^2 * s^4 * (1 + r*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).