A251693 -id:A251693 - cp's OEIS frontend

A251663 E.g.f.: exp( 3xG(x)^2 ) / G(x), where G(x) = 1 + x*G(x)^3 is the g.f. A001764.

1, 2, 11, 120, 2061, 48918, 1487151, 55188108, 2419385625, 122367255498, 7014349322739, 449405251066368, 31826192109186789, 2468711973793223070, 208159999898813165079, 18957203713618483723092, 1854424578467714146269489, 193922780991931737971748882, 21588348501840566333913576795

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 2*x + 11*x^2/2! + 120*x^3/3! + 2061*x^4/4! + 48918*x^5/5! +...
such that A(x) = exp(3*x*G(x)^2) / G(x)
where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
The e.g.f. satisfies:
A(x) = 1 + 2*x/A(x)^2 + 27*x^2/(2!*A(x)^4) + 756*x^3/(3!*A(x)^6) + 32805*x^4/(4!*A(x)^8) + 1940598*x^5/(5!*A(x)^10) + 145746783*x^6/(6!*A(x)^12) + 13286025000*x^7/(7!*A(x)^14) +...+ (n+1)*(2*n+1)^(n-2)*3^n * x^n/(n!*A(x)^(2*n)) +...

Cf. A251573, A251693, A001764.

Cf. Variants: A243953, A251664, A251665, A251666, A251667, A251668, A251669, A251670.

Table[Sum[3^k * n!/k! * Binomial[3*n-k-2, n-k] * (2*k-1)/(2*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^3 +x*O(x^n)); n!*polcoeff(exp(3*x*G^2)/G, n)}
for(n=0, 20, print1(a(n), ", "))

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

Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^2 + G'(x)/G(x).
(2) A(x) = F(x/A(x)^2) where F(x) is the e.g.f. of A251693.
(3) A(x) = Sum_{n>=0} A251693(n)*(x/A(x)^2)^n/n! where A251693(n) = (n+1) * (2*n+1)^(n-2) * 3^n.
(4) [x^n/n!] A(x)^(2*n+1) = (n+1) * (2*n+1)^(n-1) * 3^n.
a(n) = Sum_{k=0..n} 3^k * n!/k! * binomial(3*n-k-2, n-k) * (2*k-1)/(2*n-1) for n>=0.
Recurrence: 2*(2*n-1)*(9*n^2 - 30*n + 19)*a(n) = 3*(81*n^4 - 432*n^3 + 756*n^2 - 393*n - 88)*a(n-1) - 27*(9*n^2 - 12*n - 2)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 3^(3*n-3/2) * n^(n-1) / (2^(2*n-1/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

A251694 a(n) = (2n+1) (3n+1)^(n-2) 4^n.

Original entry on oeis.org

1, 3, 80, 4480, 389376, 46137344, 6939332608, 1266556600320, 272000000000000, 67204714785144832, 18780742521990414336, 5857307089452073484288, 2016866466756967373209600, 759982437118771200000000000, 311070869724989874190180941824, 137440072511222468264810285891584

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 3*x + 80*x^2/2! + 4480*x^3/3! + 389376*x^4/4! + 46137344*x^5/5! +...
such that A(x) = exp( 4*x*A(x)^3 * G(x*A(x)^3)^3 ) / G(x*A(x)^3),
where G(x) = 1 + x*G(x)^4 is the g.f. A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^3) where
F(x) = 1 + 3*x + 26*x^2/2! + 430*x^3/3! + 10872*x^4/4! + 373664*x^5/5! +...
F(x) = exp( 4*x*G(x)^3 ) / G(x) is the e.g.f. of A251664.

G. C. Greubel, Table of n, a(n) for n = 0..283

Cf. A251664, A002293.

Cf. Variants: A127670, A251693, A251695, A251696, A251697, A251698, A251699, A251700.

[ (2*n + 1)*(3*n + 1)^(n - 2)*4^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[(2*n + 1)*(3*n + 1)^(n - 2)*4^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^4 +x*O(x^n));
A = ( serreverse( x*G^3 / exp(12*x*G^3) )/x )^(1/3); n!*polcoeff(A, n)}
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) = exp( 4*x*A(x)^3 * G(x*A(x)^3)^3 ) / G(x*A(x)^3).
(2) A(x) = F(x*A(x)^3) where F(x) = exp(4*x*G(x)^3)/G(x) is the e.g.f. of A251664.
(3) A(x) = ( Series_Reversion( x*G(x)^3 / exp(12*x*G(x)^3) )/x )^(1/3).
E.g.f.: (-LambertW(-12*x)/(12*x))^(1/3) * (1 + LambertW(-12*x)/12). - Vaclav Kotesovec, Dec 07 2014

A251695 a(n) = (3n+1) (4n+1)^(n-2) 5^n.

Original entry on oeis.org

1, 4, 175, 16250, 2348125, 463050000, 115966796875, 35253537343750, 12611991884765625, 5191587030710937500, 2417311348659677734375, 1256208098030090332031250, 720779749270420907470703125, 452589644988876542822265625000, 308707218248583408960223388671875

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 4*x + 175*x^2/2! + 16250*x^3/3! + 2348125*x^4/4! + 463050000*x^5/5! +...
such that A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4),
where G(x) = 1 + x*G(x)^5 is the g.f. A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^4) where
F(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
F(x) = exp( 5*x*G(x)^4 ) / G(x) is the e.g.f. of A251665.

G. C. Greubel, Table of n, a(n) for n = 0..268

Cf. A251665, A002294.

Cf. Variants: A127670, A251693, A251694, A251696, A251697, A251698, A251699, A251700.

[(3*n + 1)*(4*n + 1)^(n - 2)*5^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[(3*n + 1)*(4*n + 1)^(n - 2)*5^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^5 +x*O(x^n));
A = ( serreverse( x*G^4 / exp(20*x*G^4) )/x )^(1/4); n!*polcoeff(A, n)}
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) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4).
(2) A(x) = F(x*A(x)^4) where F(x) = exp(5*x*G(x)^4)/G(x) is the e.g.f. of A251665.
(3) A(x) = ( Series_Reversion( x*G(x)^4 / exp(20*x*G(x)^4) )/x )^(1/4).
E.g.f.: (-LambertW(-20*x)/(20*x))^(1/4) * (1 + LambertW(-20*x)/20). - Vaclav Kotesovec, Dec 07 2014

A251696 a(n) = (4n+1) (5n+1)^(n-2) 6^n.

Original entry on oeis.org

1, 5, 324, 44928, 9716112, 2870090496, 1077194894400, 490873123897344, 263285585800098048, 162505400851637010432, 113463916253636561519616, 88423664876285081860177920, 76086820231309990402228260864, 71651521268311905104861664903168, 73298071049899905319337719679434752

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 5*x + 324*x^2/2! + 44928*x^3/3! + 9716112*x^4/4! + 2870090496*x^5/5! +...
such that A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5),
where G(x) = 1 + x*G(x)^6 is the g.f. A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^5) where
F(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...
F(x) = exp( 6*x*G(x)^5 ) / G(x) is the e.g.f. of A251666.

G. C. Greubel, Table of n, a(n) for n = 0..256

Cf. A251666, A002295.

Cf. Variants: A127670, A251693, A251694, A251695, A251697, A251698, A251699, A251700.

[(4*n + 1)*(5*n + 1)^(n - 2)*6^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(4*n + 1)*(5*n + 1)^(n - 2)*6^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^6 +x*O(x^n));
A = ( serreverse( x*G^5 / exp(30*x*G^5) )/x )^(1/5); n!*polcoeff(A, n)}
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) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5).
(2) A(x) = F(x*A(x)^5) where F(x) = exp(6*x*G(x)^5)/G(x) is the e.g.f. of A251666.
(3) A(x) = ( Series_Reversion( x*G(x)^5 / exp(30*x*G(x)^5) )/x )^(1/5).
E.g.f.: (-LambertW(-30*x)/(30*x))^(1/5) * (1 + LambertW(-30*x)/30). - Vaclav Kotesovec, Dec 07 2014

A251697 a(n) = (5n+1) (6n+1)^(n-2) 7^n.

Original entry on oeis.org

1, 6, 539, 104272, 31513125, 13018130762, 6835288192159, 4358439870247764, 3271482918202092041, 2826044644022395468750, 2761781119675422226696419, 3012587650584028093856586776, 3628565076873134344787430377389, 4783177086109789054912470697687698, 6849486554475843842876951982177734375

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 6*x + 539*x^2/2! + 104272*x^3/3! + 31513125*x^4/4! + 13018130762*x^5/5! +...
such that A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6),
where G(x) = 1 + x*G(x)^7 is the g.f. A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^6) where
F(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
F(x) = exp( 7*x*G(x)^6 ) / G(x) is the e.g.f. of A251667.

G. C. Greubel, Table of n, a(n) for n = 0..248

Cf. A251667, A002296.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251698, A251699, A251700.

[(5*n + 1)*(6*n + 1)^(n - 2)*7^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(5*n + 1)*(6*n + 1)^(n - 2)*7^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^7 +x*O(x^n));
A = ( serreverse( x*G^6 / exp(42*x*G^6) )/x )^(1/6); n!*polcoeff(A, n)}
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) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6).
(2) A(x) = F(x*A(x)^6) where F(x) = exp(7*x*G(x)^6)/G(x) is the e.g.f. of A251667.
(3) A(x) = ( Series_Reversion( x*G(x)^6 / exp(42*x*G(x)^6) )/x )^(1/6).
E.g.f.: (-LambertW(-42*x)/(42*x))^(1/6) * (1 + LambertW(-42*x)/42). - Vaclav Kotesovec, Dec 07 2014

A251698 a(n) = (6n+1) (7n+1)^(n-2) 8^n.

Original entry on oeis.org

1, 7, 832, 214016, 86118400, 47393538048, 33160072265728, 28180480000000000, 28194546272924860416, 32466269569728810844160, 42295727044150128912891904, 61505801717703291002224115712, 98762474157744880353280000000000, 173565347832317233669371533581090816, 331360760866451564310212841997955235840

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 7*x + 832*x^2/2! + 214016*x^3/3! + 86118400*x^4/4! + 47393538048*x^5/5! +...
such that A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7),
where G(x) = 1 + x*G(x)^8 is the g.f. A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^7) where
F(x) = 1 + 7*x + 146*x^2/2! + 5570*x^3/3! + 316376*x^4/4! + 24070168*x^5/5! +...
F(x) = exp( 8*x*G(x)^7 ) / G(x) is the e.g.f. of A251668.

G. C. Greubel, Table of n, a(n) for n = 0..240

Cf. A251668, A007556.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251699, A251700.

[(6*n + 1)*(7*n + 1)^(n - 2)*8^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(6*n + 1)*(7*n + 1)^(n - 2)*8^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^8 +x*O(x^n));
A = ( serreverse( x*G^7 / exp(56*x*G^7) )/x )^(1/7); n!*polcoeff(A, n)}
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) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7).
(2) A(x) = F(x*A(x)^7) where F(x) = exp(8*x*G(x)^7)/G(x) is the e.g.f. of A251668.
(3) A(x) = ( Series_Reversion( x*G(x)^7 / exp(56*x*G(x)^7) )/x )^(1/7).
E.g.f.: (-LambertW(-56*x)/(56*x))^(1/7) * (1 + LambertW(-56*x)/56). - Vaclav Kotesovec, Dec 07 2014

A251699 a(n) = (7n+1) (8n+1)^(n-2) 9^n.

Original entry on oeis.org

1, 8, 1215, 400950, 207202941, 146509780644, 131737019154363, 143893722808861650, 185052548743241765625, 273919266332571877019712, 458736814135093804224189111, 857575304752878031562956215918, 1770298011965146072399475770453365, 3999656915702652258291935606835937500

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 8*x + 1215*x^2/2! + 400950*x^3/3! + 207202941*x^4/4! + 146509780644*x^5/5! +...
such that A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8),
where G(x) = 1 + x*G(x)^9 is the g.f. A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^8) where
F(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
F(x) = exp( 9*x*G(x)^8 ) / G(x) is the e.g.f. of A251669.

G. C. Greubel, Table of n, a(n) for n = 0..235

Cf. A251669, A062994.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251700.

[(7*n + 1)*(8*n + 1)^(n - 2)*9^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(7*n + 1)*(8*n + 1)^(n - 2)*9^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n));
A = ( serreverse( x*G^8 / exp(72*x*G^8) )/x )^(1/8); n!*polcoeff(A, n)}
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) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8).
(2) A(x) = F(x*A(x)^8) where F(x) = exp(9*x*G(x)^8)/G(x) is the e.g.f. of A251669.
(3) A(x) = ( Series_Reversion( x*G(x)^8 / exp(72*x*G(x)^8) )/x )^(1/8).
E.g.f.: (-LambertW(-72*x)/(72*x))^(1/8) * (1 + LambertW(-72*x)/72). - Vaclav Kotesovec, Dec 07 2014

A251700 a(n) = (8n+1) (9n+1)^(n-2) 10^n.

Original entry on oeis.org

1, 9, 1700, 700000, 451770000, 399077600000, 448380625000000, 612032839680000000, 983672470878500000000, 1819783935144064000000000, 3809045473682732010000000000, 8900000000000000000000000000000, 22963427643543537169897000000000000, 64847222511564256853913600000000000000

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

In general, sequence a(n) = (b*n+1)*(c*n+1)^(n-2)*d^n, b > 0, c > 0, d > 0, has e.g.f. (1 + (1-b/c)*LambertW(-c*d*x)/(c+1)) * (-LambertW(-c*d*x)/(c*d*x))^(1/c). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + 9*x + 1700*x^2/2! + 700000*x^3/3! + 451770000*x^4/4! + 399077600000*x^5/5! +...
such that A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9),
where G(x) = 1 + x*G(x)^10 is the g.f. A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^9) where
F(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
F(x) = exp( 10*x*G(x)^9 ) / G(x) is the e.g.f. of A251670.

G. C. Greubel, Table of n, a(n) for n = 0..230

Cf. A251670, A059968.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251699.

[(8*n + 1)*(9*n + 1)^(n - 2)*10^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(8*n + 1)*(9*n + 1)^(n - 2)*10^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n));
A = ( serreverse( x*G^9 / exp(90*x*G^9) )/x )^(1/9); n!*polcoeff(A, n)}
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) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9).
(2) A(x) = F(x*A(x)^9) where F(x) = exp(10*x*G(x)^9)/G(x) is the e.g.f. of A251670.
(3) A(x) = ( Series_Reversion( x*G(x)^9 / exp(90*x*G(x)^9) )/x )^(1/9).
E.g.f.: (-LambertW(-90*x)/(90*x))^(1/9) * (1 + LambertW(-90*x)/90). - Vaclav Kotesovec, Dec 07 2014

cp's OEIS Frontend

A251663 E.g.f.: exp( 3*x*G(x)^2 ) / G(x), where G(x) = 1 + x*G(x)^3 is the g.f. A001764.

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A251694 a(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n.

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A251695 a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n.

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A251696 a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n.

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A251697 a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n.

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A251698 a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n.

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A251699 a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.

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A251663 E.g.f.: exp( 3xG(x)^2 ) / G(x), where G(x) = 1 + x*G(x)^3 is the g.f. A001764.

A251694 a(n) = (2n+1) (3n+1)^(n-2) 4^n.

A251695 a(n) = (3n+1) (4n+1)^(n-2) 5^n.

A251696 a(n) = (4n+1) (5n+1)^(n-2) 6^n.

A251697 a(n) = (5n+1) (6n+1)^(n-2) 7^n.

A251698 a(n) = (6n+1) (7n+1)^(n-2) 8^n.

A251699 a(n) = (7n+1) (8n+1)^(n-2) 9^n.

A251700 a(n) = (8n+1) (9n+1)^(n-2) 10^n.