cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A153296 G.f.: A(x) = F(x*G(x)^3) = F(G(x)-1) where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 29, 180, 1162, 7698, 51950, 355531, 2460224, 17178755, 120861710, 855828960, 6094211829, 43610311298, 313449094851, 2261820356684, 16379528485200, 119003715014955, 867198605427231, 6336861345197670
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 180*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 + 2974*x^5 +...
G(x)^3*A(x)^2 = 1 + 5*x + 29*x^2 + 180*x^3 + 1162*x^4 +...

Crossrefs

Cf. A000108, A001764; A153295, A153297.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(3*(n-k)+3*k,n-k)*3*k/(3*(n-k)+3*k)))}

Formula

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(3n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^2 where G(x) is the g.f. of A001764.
G.f. satisfies: A(x/F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A000108.

A153297 G.f.: A(x) = F(x*G(x)^2)^2 where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 9, 48, 276, 1656, 10212, 64190, 409218, 2637282, 17143506, 112224228, 738926064, 4889332266, 32488240779, 216664589058, 1449568426292, 9725637277248, 65417353098837, 441013558347228, 2979206654245122
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^2)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 276*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...

Crossrefs

Cf. A000108, A001764; A153296, A153298.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+2,k)*2/(2*k+2)*binomial(3*(n-k)+2*k,n-k)*2*k/(3*(n-k)+2*k)))}

Formula

a(n) = Sum_{k=0..n} C(2k+2,k)/(k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x/F(x)) = F(x*F(x))^2 where F(x) is the g.f. of A000108 (Catalan).

A153298 G.f.: A(x) = F(x*G(x)^3)^2 = F(G(x)-1)^2 where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 11, 68, 443, 2974, 20361, 141356, 991738, 7015814, 49967892, 357896120, 2575844046, 18616823352, 135051785186, 982949932092, 7175591019313, 52524480778590, 385429134781530, 2834791998208500, 20893844524709649
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^3)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...

Crossrefs

Cf. A000108, A001764; A153297, A153299.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+2,k)*2/(2*k+2)*binomial(3*(n-k)+3*k,n-k)*3*k/(3*(n-k)+3*k)))}

Formula

a(n) = Sum_{k=0..n} C(2k+2,k)/(k+1) * C(3n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x/F(x)) = F(x*F(x)^2)^2 where F(x) is the g.f. of A000108.

A153390 G.f.: A(x) = F(x*G(x))^2 where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 2, 9, 48, 278, 1696, 10736, 69886, 465019, 3149476, 21643433, 150554144, 1058101315, 7502183626, 53599160532, 385494328218, 2788827078507, 20280590381098, 148167425970522, 1087007419753186, 8004683588800899
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x))^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 278*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...

Crossrefs

Cf. A000108, A001764; A153299, A153391.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+2,k)*2/(3*k+2)*binomial(2*(n-k)+k,n-k)*k/(2*(n-k)+k)))}

Formula

a(n) = Sum_{k=0..n} C(3k+2,k)*2/(3k+2) * C(2n-k,n-k)*k/(2n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x*F(x)) = F(x*F(x)^2)^2 where F(x) is the g.f. of A001764.

A153391 G.f.: A(x) = F(x*G(x)^2) where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 1, 5, 29, 183, 1223, 8525, 61366, 453003, 3412077, 26124599, 202748728, 1591450129, 12612760009, 100790253764, 811227147197, 6570431009209, 53512143110041, 437976298197769, 3600504527707557, 29716593448484673
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 183*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 449*x^4 + 3102*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*x^4 + 5799*x^5 +...
G(x)^2*A(x)^3 = 1 + 5*x + 29*x^2 + 183*x^3 + 1223*x^4 + 8525*x^5 +...

Crossrefs

Cf. A000108, A001764; A153390, A153392.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(2*(n-k)+2*k,n-k)*2*k/(2*(n-k)+2*k)))}

Formula

a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A000108.
G.f. satisfies: A(x*F(x)) = F(F(x)-1) where F(x) is the g.f. of A001764.

A153392 G.f.: A(x) = F(x*G(x)^3) where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 1, 6, 39, 272, 2001, 15333, 121266, 983274, 8133564, 68382628, 582700485, 5021538753, 43690059657, 383263396836, 3386175566418, 30104702903914, 269125162789764, 2417709649413102, 21815252320257250, 197620659225838530
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 39*x^3 + 272*x^4+... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 90*x^4 + 297*x^5 + 1001*x^6 +...
A(x)^2 = 1 + 2*x + 13*x^2 + 90*x^3 + 658*x^4 + 5014*x^5 +...
A(x)^3 = 1 + 3*x + 21*x^2 + 154*x^3 + 1176*x^4 + 9264*x^5 +...
G(x)^3*A(x)^3 = 1 + 6*x + 39*x^2 + 272*x^3 + 2001*x^4 + 15333*x^5 +...

Crossrefs

Cf. A000108, A001764; A153391, A153393.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(2*(n-k)+3*k,n-k)*3*k/(2*(n-k)+3*k)))}

Formula

a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(2n+k,n-k)*3k/(2n+k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^3 where G(x) is the g.f. of A000108.
G.f. satisfies: A(x*F(x)) = F(x*F(x)^4) where F(x) is the g.f. of A001764.

A153393 G.f.: A(x) = F(x*G(x)^2)^2 where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 2, 11, 68, 449, 3102, 22167, 162626, 1218411, 9285888, 71778489, 561453704, 4436120129, 35354290118, 283876985742, 2294347190142, 18650560232199, 152386763938940, 1250801705584643, 10308949444236522, 85281112255921359
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^2)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 449*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...

Crossrefs

Cf. A000108, A001764; A153392, A153394.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+2,k)*2/(3*k+2)*binomial(2*(n-k)+2*k,n-k)*2*k/(2*(n-k)+2*k)))}

Formula

a(n) = Sum_{k=0..n} C(3k+2,k)*2/(3k+2) * C(2n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x*F(x)) = F(F(x)-1)^2 where F(x) is the g.f. of A001764.

A153394 G.f.: A(x) = F(x*G(x)^2)^3 where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 3, 18, 118, 813, 5799, 42470, 317637, 2416671, 18649874, 145655292, 1149199212, 9146686605, 73354982763, 592217363334, 4809250320023, 39258457746069, 321964620209940, 2651536017682988, 21919266484180533, 181820251665093357
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^2)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...

Crossrefs

Cf. A000108, A001764; A153393, A153395.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+3,k)/(k+1)*binomial(2*n,n-k)*k/n))}

Formula

a(n) = Sum_{k=0..n} C(3k+3,k)/(k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x*F(x)) = F(x*F(x)^3)^3 = F(F(x)-1)^3 where F(x) is the g.f. of A001764.

A153398 G.f.: A(x) = F(x*G(x)^2) where F(x) = G(x/F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x*G(x)) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 5, 33, 245, 1941, 16023, 136075, 1179833, 10392981, 92701411, 835271032, 7589337123, 69444928453, 639280878401, 5915683250220, 54991636090761, 513257729193329, 4807619948647095, 45177320023095160, 425766248463523359
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2009

Keywords

Examples

			G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 33*x^3 + 245*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 76*x^3 + 581*x^4 + 4702*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 130*x^3 + 1023*x^4 + 8457*x^5 +...
G(x)^2*A(x)^3 = 1 + 5*x + 33*x^2 + 245*x^3 + 1941*x^4 + 16023*x^5 +...

Crossrefs

Cf. A000108, A001764, A002293; A153397, A153399.

Programs

  • PARI
    {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(4*(n-k)+2*k,n-k)*2*k/(4*(n-k)+2*k)))}

Formula

a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(4n-2k,n-k)*k/(2n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A002293.
G.f. satisfies: A(x/F(x)) = F(x*F(x)) where F(x) is the g.f. of A001764.
G.f. satisfies: A(x/H(x)^2) = F(x) where H(x) = 1 + x*H(x)^2 is the g.f. of A000108 and F(x) is the g.f. of A001764.

A357793 a(n) = coefficient of x^n in A(x) = Sum_{n>=0} x^n*F(x)^n * (1 - x^n*F(x)^n)^n, where F(x) = 1 + x*F(x)^3 is a g.f. of A001764.

Original entry on oeis.org

1, 1, 1, 4, 14, 64, 314, 1633, 8826, 49107, 279349, 1617290, 9498099, 56445918, 338817460, 2051182532, 12509647159, 76785827812, 474000090118, 2940761033970, 18327028477625, 114677403429121, 720191795608082, 4537925593859911, 28679991910774479, 181761824439041725
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n*F(x)^n)^n / (1 - x*F(x)^2)^n, where F(x) = 1 + x*F(x)^3.

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 14*x^4 + 64*x^5 + 314*x^6 + 1633*x^7 + 8826*x^8 + 49107*x^9 + 279349*x^10 + 1617290*x^11 + 9498099*x^12 + ...
where
F(x) = 1 + x*F(x)*(1 - x*F(x)) + x^2*F(x)^2*(1 - x^2*F(x)^2) + x^3*F(x)^3*(1 - x^3*F(x)^3) + x^4*F(x)^4*(1 - x^4*F(x)^4) + ... + x^n * F(x)^n * (1 - x^n*F(x)^n)^n + ...
also,
F(x) = 1/(1 - x*F(x)) - (x*F(x))^2/(1 - x^2*F(x)^2)^2 + (x*F(x))^6/(1 - x^3*F(x)^3)^3 - (x*F(x))^12/(1 - x^4*F(x)^4)^4 + (x*F(x))^20/(1 - x^5*F(x)^4)^5 +- ... + (-1)^(n-1) * (x*F(x))^(n*(n-1)) / (1 - x^n*F(x)^n)^n + ...
Where F(x) = 1 + x*F(x)^3 begins
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + 43263*x^8 + 246675*x^9 + 1430715*x^10 + ... + A001764(n)*x^n + ...
SPECIFIC VALUES.
The radius of convergence of the power series A(x) equals 4/27.
The power series A(x) converges at x = 4/27 to
A(4/27) = 1.2311920996301390036800654138630946234233891541082821783156...
which equals the following sums:
(1) A(4/27) = Sum_{n>=0} 2^n * (9^n - 2^n)^n / 9^(n*(n+1)),
(2) A(4/27) = Sum_{n>=1} (-1)^(n-1) * 2^(n*(n-1)) * 9^n / (9^n - 2^n)^n.

Links

Crossrefs

Cf. A357792, A001764.

Programs

  • PARI
    {a(n) = my(A=1, F = (serreverse(x/(1+x + O(x^(n+2)))^3)/x)^(1/3));
A = sum(m=0,n, x^m * F^m * (1 - x^m*F^m)^m); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A=1, F = (serreverse(x/(1+x + O(x^(n+3)))^3)/x)^(1/3));
A = sum(m=1,n+1, (-1)^(m-1) * (x*F)^(m*(m-1)) / (1 - x^m*F^m)^m); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

Given F(x) = 1 + x*F(x)^3, g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} x^n * F(x)^n * (1 - x^n*F(x)^n)^n.
(2) A(x) = Sum_{n>=1} (-1)^(n-1) * (x*F(x))^(n*(n-1)) / (1 - x^n*F(x)^n)^n.
(3) A(x) = Sum_{n>=0} x^n * (1 - x^n*F(x)^n)^n / (1 - x*F(x)^2)^n.
(4) A(x) = Sum_{n>=1} (-1)^(n-1) * x^(n*(n-1)) * F(x)^(n^2) * (1 - x*F(x)^2)^n / (1 - x^n*F(x)^n)^n.
a(n) ~ c * 3^(3*n) / (n^(3/2) * 2^(2*n)), where c = 0.0403028056146458801802487899052088995113692232406693619.... - Vaclav Kotesovec, Mar 14 2023
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