cp's OEIS Frontend

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

Recurrence: 5*(n-1)*n*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(5*n + 2)*a(n) = + 576*(n-1)^2*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*a(n-2) + 288*(3*n - 5)*(3*n - 4)*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(11250*n^7 - 111375*n^6 + 440175*n^5 - 888545*n^4 + 975241*n^3 - 574177*n^2 + 165869*n - 18018)*a(n-3) + 144*(5*n - 26)*(5*n - 21)*(5*n - 19)*(5*n - 6)*(10125000*n^11 - 218700000*n^10 + 2075585625*n^9 - 11378954250*n^8 + 39836289925*n^7 - 92894908470*n^6 + 145953551806*n^5 - 152681445300*n^4 + 102505633480*n^3 - 41086190160*n^2 + 8557182144*n - 670602240)*a(n-4) + 72*(5*n - 26)*(5*n - 11)*(5*n - 6)*(5*n - 4)*(20250000*n^11 - 569025000*n^10 + 7025658750*n^9 - 50083579125*n^8 + 227686012400*n^7 - 687547140050*n^6 + 1391232445598*n^5 - 1854143517725*n^4 + 1550931293540*n^3 - 737424345140*n^2 + 162058858752*n - 10360465920)*a(n-5) + 144*(5*n - 16)*(5*n - 11)*(5*n - 9)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(202500*n^9 - 5953500*n^8 + 74924775*n^7 - 526434885*n^6 + 2255339082*n^5 - 6025054075*n^4 + 9796892735*n^3 - 8893818500*n^2 + 3545754268*n - 142331280)*a(n-6) + 72*n*(2*n - 9)*(3*n - 17)*(3*n - 10)*(5*n - 21)*(5*n - 16)*(5*n - 14)*(5*n - 11)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*(6*n - 41)*(6*n - 13)*a(n-7). - Vaclav Kotesovec, Nov 18 2017

a(n) ~ sqrt((1 + 2*r*s^6)/(15*Pi)) / (2*s^2 * n^(3/2) * r^(n + 1/2)), where r = 0.1734895129039028676461340698295316044509963479582... and s = 1.408187415484683441175360883795437925341195617549... are roots of the system of equations 1 + r + r^2*s^6 = s, 6*r^2*s^5 = 1. - Vaclav Kotesovec, Nov 18 2017

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 A137953.

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

Recurrence: 5*n*(5*n - 4)*(5*n - 3)*(5*n - 1)*(5*n + 3)*(8845200*n^11 - 252428400*n^10 + 3221192232*n^9 - 24137808840*n^8 + 117463352781*n^7 - 387964460127*n^6 + 882822962553*n^5 - 1374856808005*n^4 + 1422227015434*n^3 - 915895407668*n^2 + 320324023880*n - 42693386400)*a(n) = - 360*(5*n - 2)*(5670000*n^13 - 63714600*n^12 - 1032645960*n^11 + 24848001198*n^10 - 218480624507*n^9 + 1101741928166*n^8 - 3582401014336*n^7 + 7865579681092*n^6 - 11836392808433*n^5 + 12130520012664*n^4 - 8236278842764*n^3 + 3497924862840*n^2 - 827741189520*n + 81691545600)*a(n-1) + 180*(2653560000*n^16 - 86342760000*n^15 + 1284348733200*n^14 - 11544882534000*n^13 + 69915022739748*n^12 - 301277354913324*n^11 + 951521048997123*n^10 - 2235356609743737*n^9 + 3921814538564296*n^8 - 5108337175422974*n^7 + 4854490688899951*n^6 - 3250616687965913*n^5 + 1431302003002666*n^4 - 349408874612852*n^3 + 16089460853736*n^2 + 12240998632800*n - 2031289747200)*a(n-2) + 72*(17672709600*n^16 - 601551846000*n^15 + 9383367519936*n^14 - 88661500185240*n^13 + 565349613141438*n^12 - 2565633937621131*n^11 + 8513410651166583*n^10 - 20875837005697545*n^9 + 37705724089968084*n^8 - 49181218885648923*n^7 + 44098626888119141*n^6 - 23771481353637565*n^5 + 3467317211974378*n^4 + 4824415011450004*n^3 - 3654086377331160*n^2 + 1070168332564800*n - 116760296016000)*a(n-3) + 144*(8597534400*n^16 - 305543145600*n^15 + 4975684360704*n^14 - 49077873815616*n^13 + 326509076764188*n^12 - 1543742190898488*n^11 + 5321067950386782*n^10 - 13479709842928188*n^9 + 24903384308348709*n^8 - 32579354322085314*n^7 + 27941366702438094*n^6 - 11913061039189846*n^5 - 3157851308946897*n^4 + 7647346836930652*n^3 - 4534021704525180*n^2 + 1245319349576400*n - 132684717816000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 10)*(6*n - 25)*(6*n - 23)*(8845200*n^11 - 155131200*n^10 + 1183394232*n^9 - 5046898752*n^8 + 12951310413*n^7 - 19922972292*n^6 + 16394061984*n^5 - 2858995378*n^4 - 7011543813*n^3 + 6369403462*n^2 - 2180183136*n + 267092640)*a(n-5). - Vaclav Kotesovec, Nov 18 2017

a(n) ~ sqrt((1 + 4*r*s^3 + 3*r^2*s^6) / (3*Pi*s*(2 + 5*r*s^3))) / (2*n^(3/2) * r^(n + 1/2)), where r = 0.1898739884773465982357897900946346962414966313829... and s = 1.607584028097173055359903977736399386285943742600... are roots of the system of equations 1 + r*(1 + r*s^3)^2 = s, 6*r^2*s^2*(1 + r*s^3) = 1. - Vaclav Kotesovec, Nov 18 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

G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137967.

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

a(n) ~ sqrt(6*s*(1-s)*(2-3*s) / ((44*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.572098697306844482137442690518486437859864764710... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^6, 12 * r^2 * s * (1 + r*s^2)^5 = 1. - Vaclav Kotesovec, Nov 22 2017

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

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A137967.