A243953 -id:A243953 - cp's OEIS frontend

A251664 E.g.f.: exp(4xG(x)^3) / G(x) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 3, 26, 430, 10872, 373664, 16295152, 862486944, 53729041280, 3851892172288, 312411790027776, 28284076403710208, 2827642792215049216, 309396856974126428160, 36777992050266076762112, 4719560392385576181243904, 650284066459536965937364992, 95752333835299098922624876544, 15005473998204120386383308390400

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 3*x + 26*x^2/2! + 430*x^3/3! + 10872*x^4/4! + 373664*x^5/5! +...
such that A(x) = exp(4*x*G(x)^3) / G(x)
where G(x) = 1 + x*G(x)^4 is the g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Cf. A251574, A251694, A002293.

Cf. Variants: A243953, A251663, A251665, A251666, A251667, A251668, A251669, A251670.

Table[Sum[4^k * n!/k! * Binomial[4*n-k-2,n-k] * (3*k-1)/(3*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G=1+x*G^4 +x*O(x^n)); n!*polcoeff(exp(4*x*G^3)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 4^k * n!/k! * binomial(4*n-k-2,n-k) * (3*k-1)/(3*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^3 + 2*G'(x)/G(x).
(2) A(x) = F(x/A(x)^3) where F(x) is the e.g.f. of A251694.
(3) A(x) = Sum_{n>=0} A251694(n)*(x/A(x)^3)^n/n! where A251694(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n.
(4) [x^n/n!] A(x)^(3*n+1) = (2*n+1) * (3*n+1)^(n-1) * 4^n.
a(n) = Sum_{k=0..n} 4^k * n!/k! * binomial(4*n-k-2,n-k) * (3*k-1)/(3*n-1) for n>=0.
Recurrence: 3*(3*n-2)*(3*n-1)*(64*n^3 - 344*n^2 + 598*n - 315)*a(n) = 8*(2048*n^6 - 16128*n^5 + 51136*n^4 - 82160*n^3 + 67332*n^2 - 22212*n - 855)*a(n-1) + 256*(64*n^3 - 152*n^2 + 102*n + 3)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 2^(8*n-2) / 3^(3*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251665 E.g.f.: exp(5xG(x)^4) / G(x) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

Original entry on oeis.org

1, 4, 47, 1034, 34349, 1540480, 87311275, 5991370390, 483100288985, 44778459212540, 4691799973171175, 548418557098305250, 70754785462138421125, 9987462340422594014200, 1531136319790275407365475, 253347224928445454055920750, 45001449932636667231257800625, 8541130421294458307989700672500

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
such that A(x) = exp(5*x*G(x)^4) / G(x)
where G(x) = 1 + x*G(x)^5 is the g.f. of A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...

Cf. A251575, A251695, A002294.

Cf. Variants: A243953, A251663, A251664, A251666, A251667, A251668, A251669, A251670.

Table[Sum[5^k * n!/k! * Binomial[5*n-k-2,n-k] * (4*k-1)/(4*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G=1+x*G^5 +x*O(x^n)); n!*polcoeff(exp(5*x*G^4)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 5^k * n!/k! * binomial(5*n-k-2,n-k) * (4*k-1)/(4*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^4 + 3*G'(x)/G(x).
(2) A(x) = F(x/A(x)^4) where F(x) is the e.g.f. of A251695.
(3) A(x) = Sum_{n>=0} A251695(n)*(x/A(x)^4)^n/n! where A251695(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n .
(4) [x^n/n!] A(x)^(4*n+1) = (3*n+1) * (4*n+1)^(n-1) * 5^n.
a(n) = Sum_{k=0..n} 5^k * n!/k! * binomial(5*n-k-2,n-k) * (4*k-1)/(4*n-1) for n>=0.
Recurrence: 8*(2*n-1)*(4*n-3)*(4*n-1)*(1875*n^4 - 13375*n^3 + 35700*n^2 - 41905*n + 17681)*a(n) = 5*(1171875*n^8 - 11875000*n^7 + 51765625*n^6 - 126596875*n^5 + 189126875*n^4 - 174442875*n^3 + 93137550*n^2 - 22362645*n - 233856)*a(n-1) - 3125*(1875*n^4 - 5875*n^3 + 6825*n^2 - 3130*n - 24)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 3 * 5^(5*n-3/2) / 2^(8*n-1) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251666 E.g.f.: exp(6xG(x)^5) / G(x) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

Original entry on oeis.org

1, 5, 74, 2028, 83352, 4607496, 321156000, 27064420704, 2677510124928, 304299947999232, 39075730095810816, 5595805388119057920, 884245579070535235584, 152843879008651568329728, 28688663318934190485491712, 5811091829207760774331662336, 1263471121829937070180445552640

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...
such that A(x) = exp(6*x*G(x)^5) / G(x)
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...

Cf. A251576, A251696, A002295.

Cf. Variants: A243953, A251663, A251664, A251665, A251667, A251668, A251669, A251670.

Table[Sum[6^k * n!/k! * Binomial[6*n-k-2,n-k] * (5*k-1)/(5*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G=1+x*G^6 +x*O(x^n)); n!*polcoeff(exp(6*x*G^5)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 6^k * n!/k! * binomial(6*n-k-2,n-k) * (5*k-1)/(5*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^5 + 4*G'(x)/G(x).
(2) A(x) = F(x/A(x)^5) where F(x) is the e.g.f. of A251696.
(3) A(x) = Sum_{n>=0} A251696(n)*(x/A(x)^5)^n/n! where A251696(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n .
(4) [x^n/n!] A(x)^(5*n+1) = (4*n+1) * (5*n+1)^(n-1) * 6^n .
a(n) = Sum_{k=0..n} 6^k * n!/k! * binomial(6*n-k-2,n-k) * (5*k-1)/(5*n-1) for n>=0.
Recurrence: 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*(1296*n^5 - 11394*n^4 + 40230*n^3 - 71274*n^2 + 63110*n - 21963)*a(n) = 144*(419904*n^10 - 5161320*n^9 + 28223964*n^8 - 90513612*n^7 + 188713962*n^6 - 267339204*n^5 + 259905051*n^4 - 169257762*n^3 + 67929146*n^2 - 12957136*n - 43050)*a(n-1) + 46656*(1296*n^5 - 4914*n^4 + 7614*n^3 - 5988*n^2 + 2156*n + 5)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 4 * 6^(6*n-3/2) / 5^(5*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251667 E.g.f.: exp(7xG(x)^6) / G(x) where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

Original entry on oeis.org

1, 6, 107, 3508, 171741, 11280842, 933014767, 93212094024, 10925496633401, 1470493880790382, 223555405538724819, 37892802280129883324, 7086076189702624109653, 1449303152891376476830962, 321848482510755456019058519, 77124029495405859198280522768

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
such that A(x) = exp(7*x*G(x)^6) / G(x)
where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...

Cf. A251577, A251697, A002295.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251668, A251669, A251670.

Table[Sum[7^k * n!/k! * Binomial[7*n-k-2,n-k] * (6*k-1)/(6*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G=1+x*G^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^6 + 5*G'(x)/G(x).
(2) A(x) = F(x/A(x)^6) where F(x) is the e.g.f. of A251697.
(3) A(x) = Sum_{n>=0} A251697(n)*(x/A(x)^6)^n/n! where A251697(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n .
(4) [x^n/n!] A(x)^(6*n+1) = (5*n+1) * (6*n+1)^(n-1) * 7^n .
a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) for n>=0.
Recurrence: 72*(2*n-1)*(3*n-2)*(3*n-1)*(6*n-5)*(6*n-1)*(588245*n^6 - 6117748*n^5 + 26651100*n^4 - 62321728*n^3 + 82554122*n^2 - 58646294*n + 17291583)*a(n) = 7*(69206436005*n^12 - 996572678472*n^11 + 6516703994430*n^10 - 25624338676965*n^9 + 67604945463195*n^8 - 126360374558838*n^7 + 171960790012102*n^6 - 171911061779835*n^5 + 125050872537045*n^4 - 63802357502870*n^3 + 20814954345360*n^2 - 3329274812661*n - 3763584000)*a(n-1) - 823543*(588245*n^6 - 2588278*n^5 + 4886035*n^4 - 5129908*n^3 + 3141733*n^2 - 958104*n - 720)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 5 * 7^(7*n-3/2) / 6^(6*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251668 E.g.f.: exp(8xG(x)^7) / G(x) where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.

Original entry on oeis.org

1, 7, 146, 5570, 316376, 24070168, 2303562256, 266121810928, 36054510934400, 5607388438811648, 984840629002206464, 192818670654633123328, 41644201910970978233344, 9836055425319263031070720, 2522269785922810486307846144, 697878768774876825573221076992, 207239855326220163290204654895104

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 7*x + 146*x^2/2! + 5570*x^3/3! + 316376*x^4/4! + 24070168*x^5/5! +...
such that A(x) = exp(8*x*G(x)^7) / G(x)
where G(x) = 1 + x*G(x)^8 is the g.f. of A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...

Cf. A251578, A251698, A007556.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251667, A251669, A251670.

Table[Sum[8^k * n!/k! * Binomial[8*n-k-2,n-k] * (7*k-1)/(7*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G=1+x*G^8 +x*O(x^n)); n!*polcoeff(exp(8*x*G^7)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 8^k * n!/k! * binomial(8*n-k-2,n-k) * (7*k-1)/(7*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^8 be the g.f. of A007556, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^7 + 6*G'(x)/G(x).
(2) A(x) = F(x/A(x)^7) where F(x) is the e.g.f. of A251698.
(3) A(x) = Sum_{n>=0} A251698(n)*(x/A(x)^7)^n/n! where A251698(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n.
(4) [x^n/n!] A(x)^(7*n+1) = (6*n+1) * (7*n+1)^(n-1) * 8^n.
a(n) = Sum_{k=0..n} 8^k * n!/k! * binomial(8*n-k-2,n-k) * (7*k-1)/(7*n-1) for n>=0.
Recurrence: 7*(7*n-6)*(7*n-5)*(7*n-4)*(7*n-3)*(7*n-2)*(7*n-1)*(786432*n^7 - 9420800*n^6 + 48617472*n^5 - 140311296*n^4 + 244933664*n^3 - 258931936*n^2 + 153488834*n - 39162055)*a(n) = 128*(103079215104*n^14 - 1698659565568*n^13 + 12890002161664*n^12 - 59777925251072*n^11 + 189611502469120*n^10 - 436161486520320*n^9 + 752457969500160*n^8 - 992208595165184*n^7 + 1008016614120448*n^6 - 786554741352448*n^5 + 462978217009024*n^4 - 196807984954976*n^3 + 54746044352828*n^2 - 7603476442560*n - 3022294275)*a(n-1) + 16777216*(786432*n^7 - 3915776*n^6 + 8607744*n^5 - 11010816*n^4 + 8972320*n^3 - 4620928*n^2 + 1248354*n + 315)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 6 * 8^(8*n-3/2) / 7^(7*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251669 E.g.f.: exp(9xG(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

Original entry on oeis.org

1, 8, 191, 8310, 537117, 46444164, 5047987707, 662002733394, 101779688986425, 17959176833948928, 3578033935192224951, 794559576204365478318, 194620831940208238831701, 52129134740350115227721340, 15158273263608217360939225587, 4755712518628181890216523759754

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...

Cf. A251579, A251699, A062994.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251667, A251668, A251670.

Table[Sum[9^k * n!/k! * Binomial[9*n-k-2,n-k] * (8*k-1)/(8*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^8 + 7*G'(x)/G(x).
(2) A(x) = F(x/A(x)^8) where F(x) is the e.g.f. of A251699.
(3) A(x) = Sum_{n>=0} A251699(n)*(x/A(x)^8)^n/n! where A251699(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.
(4) [x^n/n!] A(x)^(8*n+1) = (7*n+1) * (8*n+1)^(n-1) * 9^n.
a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) for n>=0.
Recurrence: 128*(2*n-1)*(4*n-3)*(4*n-1)*(8*n-7)*(8*n-5)*(8*n-3)*(8*n-1)*(33480783*n^8 - 453319173*n^7 + 2697889761*n^6 - 9230277240*n^5 + 19886167926*n^4 - 27672715746*n^3 + 24328423881*n^2 - 12365760717*n + 2776106045)*a(n) = 81*(160137547184727*n^16 - 2968899287488272*n^15 + 25604779347830979*n^14 - 136506824772659775*n^13 + 504285657127489314*n^12 - 1371500076773316825*n^11 + 2847804013092225933*n^10 - 4619534029925962572*n^9 + 5937710241656343834*n^8 - 6090889132598477481*n^7 + 4986522977501530773*n^6 - 3228624422259256476*n^5 + 1615386846720554091*n^4 - 595058403096826425*n^3 + 145565831993332122*n^2 - 17972427186502245*n - 2554359808000)*a(n-1) - 387420489*(33480783*n^8 - 185472909*n^7 + 462117474*n^6 - 687717459*n^5 + 680611896*n^4 - 464268429*n^3 + 210617505*n^2 - 51824070*n - 4480)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 7 * 3^(18*n-3) / 8^(8*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251670 E.g.f.: exp(10xG(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

Original entry on oeis.org

1, 9, 242, 11824, 856824, 82986080, 10097121280, 1481787433920, 254874712419200, 50305519571800960, 11209381628379724800, 2783746998856794752000, 762476362390276346060800, 228363072063685762536960000, 74247696727054926125971251200, 26044746725090717967744412672000

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

In general, sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-2,n-k) * ((m-1)*k-1)/((m-1)*n-1), m>2, is asymptotic to (m-2) * m^(m*n-3/2) / (m-1)^((m-1)*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)
where G(x) = 1 + x*G(x)^10 is the g.f. of A059958:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...

Cf. A251580, A251700, A059968.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251667, A251668, A251669.

Table[Sum[10^k * n!/k! * Binomial[10*n-k-2,n-k] * (9*k-1)/(9*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n)); n!*polcoeff(exp(10*x*G^9)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9 + 8*G'(x)/G(x).
(2) A(x) = F(x/A(x)^9) where F(x) is the e.g.f. of A251700.
(3) A(x) = Sum_{n>=0} A251700(n)*(x/A(x)^9)^n/n! where A251700(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n.
(4) [x^n/n!] A(x)^(9*n+1) = (8*n+1) * (9*n+1)^(n-1) * 10^n.
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-2,n-k) * (9*k-1)/(9*n-1) for n>=0.
Recurrence: 81*(3*n-2)*(3*n-1)*(9*n-8)*(9*n-7)*(9*n-5)*(9*n-4)*(9*n-2)*(9*n-1)*(100000000*n^9 - 1508750000*n^8 + 10158500000*n^7 - 40108637500*n^6 + 102477510000*n^5 - 175985889125*n^4 + 203494963150*n^3 - 153061617555*n^2 + 68057955478*n - 13624029912)*a(n) = 800*(1250000000000000*n^18 - 25734375000000000*n^17 + 248379687500000000*n^16 - 1494668125000000000*n^15 + 6291920187500000000*n^14 - 19707236445312500000*n^13 + 47696214907031250000*n^12 - 91443867836531250000*n^11 + 141240231848528125000*n^10 - 177729148289358906250*n^9 + 183386452781820390625*n^8 - 155416253373710737500*n^7 + 107706559814898413750*n^6 - 60246014246053412750*n^5 + 26474457002621149925*n^4 - 8675686414409435660*n^3 + 1905677176596950796*n^2 - 212632849946745072*n - 10904042717568)*a(n-1) + 10000000000*(100000000*n^9 - 608750000*n^8 + 1688500000*n^7 - 2844137500*n^6 + 3264185000*n^5 - 2692901625*n^4 + 1611256650*n^3 - 663025355*n^2 + 151278318*n + 4536)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 8 * 10^(10*n-3/2) / 3^(18*n-1) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A213507 E.g.f.: exp( Sum_{n>=1} A000108(n)x^n/n ), where A000108(n) = binomial(2n,n)/(n+1) forms the Catalan numbers.

Original entry on oeis.org

1, 1, 3, 17, 149, 1809, 28399, 550297, 12732873, 343231361, 10576764251, 367054970721, 14173669352413, 602974492511377, 28027436035348359, 1413479599558432169, 76879014760731439889, 4486205132570631391617, 279595430611791210216883, 18536284947404377562405041

Offset: 0

Paul D. Hanna, Jun 12 2012

Compare to: exp( Sum_{n>=1} (n+1)/2*A000108(n)*x^n/n ) = Sum_{n>=0} A000108(n)*x^n.

Conjecture: a(n) is not divisible by {2,5,13,29,31,37,41,43,53,71,97}.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 149*x^4/4! + 1809*x^5/5! + ...
such that
log(A(x)) = x + 2*x^2/2 + 5*x^3/3 + 14*x^4/4 + 42*x^5/5 + 132*x^6/6 + 429*x^7/7 + 1430*x^8/8 + ... + A000108(n)*x^n/n + ...

Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See p. 7.

Cf. A000108, A000169, A243953.

With[{nn=20},CoefficientList[Series[(4Exp[1-2/(Sqrt[1-4x]+1)])/ (Sqrt[ 1-4x]+1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 18 2016 *)

{a(n)=n!*polcoeff(exp(sum(m=1,n,binomial(2*m,m)/(m+1)*x^m/m)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: (4*exp(1-2/(sqrt(1-4*x)+1)))/(sqrt(1-4*x)+1)^2. - Vladimir Kruchinin, Mar 22 2013
a(n) ~ 2^(2*n+5/2)*n^(n-2)/exp(n+1). - Vaclav Kotesovec, Sep 24 2013
From Peter Bala, Nov 21 2016: (Start)
Conjectural e.g.f.: 1/x * Series_Reversion ( x/(1 + F(x)) ), where F(x) = x + Sum_{n >= 2} (n - 1)^(n - 2)*x^n/n! = x + x^2/2! + 2*x^3/3! + 3^2*x^4/4! + 4^3*x^5/5! + 5^4*x^6/6! + ...; that is, dF/dx = 1 - LambertW(-x) = 1 + Euler's tree function T(x). See A000169.
The conjecture is equivalent to the result: Catalan(n) = [x^n] (1 + F(x))^n = [x^n] (2*x + x*T(x) + x/T(x))^n. (End)

A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2k,k)x^k/(k + 1)!).

Original entry on oeis.org

1, 1, 3, 12, 59, 343, 2295, 17307, 144751, 1326377, 13189945, 141271298, 1619488645, 19766050827, 255693112641, 3492065507376, 50180426293255, 756444290843433, 11930511611596861, 196404976143077964, 3367697323914503113, 60029614473492823771, 1110430594720934758781

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Exponential transform of A000108.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 12*x^3/3! + 59*x^4/4! + 343*x^5/5! + 2295*x^6/6! + 17307*x^7/7! + ...

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A001517, A002212, A007317, A080893, A213507, A243953, A302197.

a:=series(exp(add(binomial(2*k,k)*x^k/(k+1)!,k=1..100)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Exp[Sum[CatalanNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[CatalanNumber[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} A000108(k)*x^k/k!).

E.g.f.: exp(exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1).

cp's OEIS Frontend

A251664 E.g.f.: exp(4*x*G(x)^3) / G(x) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A251665 E.g.f.: exp(5*x*G(x)^4) / G(x) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A251666 E.g.f.: exp(6*x*G(x)^5) / G(x) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A251667 E.g.f.: exp(7*x*G(x)^6) / G(x) where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A251668 E.g.f.: exp(8*x*G(x)^7) / G(x) where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A251669 E.g.f.: exp(9*x*G(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A251670 E.g.f.: exp(10*x*G(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

A251664 E.g.f.: exp(4xG(x)^3) / G(x) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A251665 E.g.f.: exp(5xG(x)^4) / G(x) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

A251666 E.g.f.: exp(6xG(x)^5) / G(x) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

A251667 E.g.f.: exp(7xG(x)^6) / G(x) where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

A251668 E.g.f.: exp(8xG(x)^7) / G(x) where G(x) = 1 + x*G(x)^8 is the g.f. of A007556.

A251669 E.g.f.: exp(9xG(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

A251670 E.g.f.: exp(10xG(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

A213507 E.g.f.: exp( Sum_{n>=1} A000108(n)x^n/n ), where A000108(n) = binomial(2n,n)/(n+1) forms the Catalan numbers.

A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2k,k)x^k/(k + 1)!).