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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A359714 Central terms of triangle A359670; a(n) = A359670(2*n,n) for n >= 0.

Original entry on oeis.org

1, 6, 68, 970, 15627, 271698, 4980320, 94919382, 1864060550, 37486601966, 768542230128, 16010270917186, 338044149765168, 7220000851821450, 155743662496011552, 3388779105788095886, 74299386925266352272, 1640069094618726916032, 36421678762652448251540
Offset: 0

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Author

Paul D. Hanna, Jan 17 2023

Keywords

Comments

The g.f. G(x,y) of triangle A359670 satisfies: G(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*G(x,y) + x^n)^n].

Links

Crossrefs

Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,2*n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^(2*n)) )^m ) );
polcoeff( polcoeff( A,2*n,x),n,y)}
for(n=0,25, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,2*n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(m=-#A,#A, (-1)^m * x^m * (y*Ser(A) + x^(m-1))^(m+1) )/(-y),#A-1,x) ); polcoeff( A[2*n+1],n,y)}
for(n=0,25, print1(a(n),", "))

A359715 Column 2 of triangle A359670; a(n) = A359670(n+2,2) for n >= 0.

Original entry on oeis.org

1, 12, 68, 284, 998, 3092, 8724, 22904, 56679, 133516, 301664, 657368, 1387854, 2849168, 5704476, 11166464, 21415632, 40312176, 74593476, 135864792, 243872632, 431835140, 755039948, 1304589104, 2229192801, 3769452152, 6311385252, 10469412968, 17214152072
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Comments

The g.f. G(x,y) of triangle A359670 satisfies: G(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*G(x,y) + x^n)^n].

Crossrefs

Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n+2,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^(n+2)) )^m ) );
polcoeff( polcoeff( A,n+2,x),2,y)}
for(n=0,30, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,n+2, A = concat(A,0);
A[#A] = polcoeff(-y + sum(m=-#A,#A, (-1)^m * x^m * (y*Ser(A) + x^(m-1))^(m+1) )/(-y),#A-1,x) ); polcoeff( A[n+3],2,y)}
for(n=0,30, print1(a(n),", "))

A359718 Column 3 of triangle A359670; a(n) = A359670(n+3,3) for n >= 0.

Original entry on oeis.org

1, 20, 170, 970, 4410, 17172, 59545, 188700, 556085, 1542640, 4065868, 10253720, 24880705, 58351000, 132750390, 293867786, 634623035, 1339924290, 2771178885, 5623152080, 11211087225, 21989506510, 42478375740, 80897833810, 152022961870, 282119268256, 517394696690
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Comments

The g.f. G(x,y) of triangle A359670 satisfies: G(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*G(x,y) + x^n)^n].

Crossrefs

Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n+3,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^(n+3)) )^m ) );
polcoeff( polcoeff( A,n+3,x),3,y)}
for(n=0,30, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,n+3, A = concat(A,0);
A[#A] = polcoeff(-y + sum(m=-#A,#A, (-1)^m * x^m * (y*Ser(A) + x^(m-1))^(m+1) )/(-y),#A-1,x) ); polcoeff( A[n+4],3,y)}
for(n=0,30, print1(a(n),", "))

A359712 a(n) = coefficient of x^n in A(x) such that 2 = Sum_{n=-oo..+oo} (-x)^n * (2*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 4, 20, 106, 586, 3356, 19728, 118382, 722208, 4466050, 27931600, 176371300, 1122867012, 7199842666, 46454345844, 301384205640, 1964899532794, 12866563846920, 84585757496444, 558060746899684, 3693810227983576, 24521903234307786, 163234951757526400
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 20*x^2 + 106*x^3 + 586*x^4 + 3356*x^5 + 19728*x^6 + 118382*x^7 + 722208*x^8 + 4466050*x^9 + 27931600*x^10 +  ...

Links

Crossrefs

Cf. A359670, A359711, A359713, A363104, A363105, A361778.

Programs

  • PARI
    {a(n) = my(A=1,y=2); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1],y=2); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(n-1))^(n+1).
(2) 2*x = Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^(n+1).
(3) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n-1).
(4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^n ].
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^(n+1))^n ].
From Paul D. Hanna, May 12 2023: (Start)
(6) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (2*A(x) + x^n)^n.
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (2*A(x) + x^n)^n ].
(8) 2*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^n. (End)
a(n) = Sum_{k=0..n} A359670(n,k)*2^k for n >= 0.

A359711 a(n) = coefficient of x^n in A(x) such that 1 = Sum_{n=-oo..+oo} (-x)^n * (A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 3, 11, 42, 165, 671, 2795, 11877, 51286, 224413, 992924, 4434833, 19969030, 90550829, 413148619, 1895338362, 8737219074, 40452543831, 188025758635, 877055405522, 4104269624748, 19262955163275, 90652992751518, 427681283728070, 2022341915324936, 9583224591208298
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Comments

Row sums of triangle A359670.

Examples

			G.f.: A(x) = 1 + 3*x + 11*x^2 + 42*x^3 + 165*x^4 + 671*x^5 + 2795*x^6 + 11877*x^7 + 51286*x^8 + 224413*x^9 + 992924*x^10 + ...

Links

Crossrefs

Cf. A359670, A359712, A359713, A363104, A363105.
Cf. A363142, A363143, A363144.

Programs

  • PARI
    {a(n) = my(A=1,y=1); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1],y=1); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(n-1))^(n+1).
(2) x = Sum_{n=-oo..+oo} (-1)^n * (x*A(x) + x^n)^(n+1).
(3) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+1))^(n-1).
(4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (x*A(x) + x^n)^n ].
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^(n+1))^n ].
From Paul D. Hanna, May 18 2023: (Start)
(6) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x) + x^n)^n.
(7) A(x) = -1 / [Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^n ].
(8) x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^n)^n. (End)
a(n) = Sum_{k=0..n} A359670(n,k) for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 5.008723344615566939692217... and c = 4.45330627132612826203... - Vaclav Kotesovec, Mar 14 2023

A363142 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).

Original entry on oeis.org

1, 1, 3, 7, 17, 42, 107, 275, 715, 1884, 5009, 13421, 36224, 98382, 268657, 737244, 2032035, 5622938, 15615186, 43505382, 121570407, 340639265, 956861955, 2694064938, 7601455079, 21490621769, 60870280259, 172707869088, 490818655346, 1396973741672, 3981748142925
Offset: 0

Views

Author

Paul D. Hanna, May 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 42*x^5 + 107*x^6 + 275*x^7 + 715*x^8 + 1884*x^9 + 5009*x^10 + 13421*x^11 + 36224*x^12 + ...

Links

Crossrefs

Cf. A359711, A363143, A363144, A363182.
Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(2*m-1))^(m+1) ),#A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023

A359713 a(n) = coefficient of x^n in A(x) such that 3 = Sum_{n=-oo..+oo} (-x)^n * (3*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 5, 31, 206, 1433, 10329, 76459, 577855, 4440538, 34591555, 272545144, 2168118299, 17390330046, 140486973983, 1142036572271, 9335129425718, 76681549612006, 632655728172281, 5240339959916895, 43561574812700958, 363294379940353624, 3038799803831856805
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 31*x^2 + 206*x^3 + 1433*x^4 + 10329*x^5 + 76459*x^6 + 577855*x^7 + 4440538*x^8 + 34591555*x^9 + 272545144*x^10 + ...

Links

Crossrefs

Cf. A359670, A359711, A359712, A363104, A363105.

Programs

  • PARI
    {a(n) = my(A=1,y=3); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1],y=3); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(n-1))^(n+1).
(2) 3*x = Sum_{n=-oo..+oo} (-1)^n * (3*x*A(x) + x^n)^(n+1).
(3) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+1))^(n-1).
(4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (3*x*A(x) + x^n)^n ].
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^(n+1))^n ].
a(n) = Sum_{k=0..n} A359670(n,k)*3^k for n >= 0.

A363104 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 6, 44, 348, 2886, 24800, 218888, 1972572, 18075100, 167900506, 1577467760, 14963979584, 143124912880, 1378756186748, 13365212659144, 130274948580864, 1276075285222662, 12554452588117632, 124003727286837484, 1229203475053859456, 12224294019862383720
Offset: 0

Views

Author

Paul D. Hanna, May 21 2023

Keywords

Comments

Conjecture: g.f. A(x) == theta_3(x) (mod 4); a(n) == 2 (mod 4) iff n is a nonzero square and a(n) == 0 (mod 4) iff n is nonsquare.

Examples

			G.f.: A(x) = 1 + 6*x + 44*x^2 + 348*x^3 + 2886*x^4 + 24800*x^5 + 218888*x^6 + 1972572*x^7 + 18075100*x^8 + 167900506*x^9 + 1577467760*x^10 + ...

Links

Crossrefs

Cf. A359670, A359711, A359712, A359713, A363105.
Cf. A363184.

Programs

  • PARI
    {a(n) = my(A=1, y=4); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=[1], y=4); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(n-1))^(n+1).
(2) 4 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (4*A(x) + x^n)^n.
(3) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^(n-1).
(4) 4*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^(n+1).
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(n-1))^n ].
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (4*A(x) + x^n)^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^(n+1))^n ].
(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^n)^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^n.
a(n) = Sum_{k=0..n} A359670(n,k) * 4^k for n >= 0.

A363105 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-x)^n * (5*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 7, 59, 538, 5149, 51059, 520035, 5407889, 57181230, 612910369, 6644662132, 72731584789, 802696690614, 8922392225233, 99798739026795, 1122441028044882, 12686176392341722, 144013323190860339, 1641303449002365323, 18772674107796041770, 215413772477355781876
Offset: 0

Views

Author

Paul D. Hanna, May 21 2023

Keywords

Examples

			G.f.: A(x) = 1 + 7*x + 59*x^2 + 538*x^3 + 5149*x^4 + 51059*x^5 + 520035*x^6 + 5407889*x^7 + 57181230*x^8 + 612910369*x^9 + 6644662132*x^10 + ...

Links

Crossrefs

Cf. A359670, A359711, A359712, A359713, A363104.
Cf. A363185.

Programs

  • PARI
    {a(n) = my(A=1, y=5); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=[1], y=5); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^(n+1).
(2) 5 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (5*A(x) + x^n)^n.
(3) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n-1).
(4) 5*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n+1).
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^n ].
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (5*A(x) + x^n)^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^(n+1))^n ].
(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (5*A(x) + x^n)^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^n.
a(n) = Sum_{k=0..n} A359670(n,k) * 5^k for n >= 0.

A363182 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).

Original entry on oeis.org

1, 2, 6, 20, 68, 234, 824, 2956, 10750, 39540, 146864, 550096, 2075432, 7880046, 30086704, 115445028, 444941028, 1721720032, 6686357238, 26051961396, 101810056296, 398962013908, 1567354966200, 6171824148252, 24355381522328, 96304034538898, 381506619687824
Offset: 0

Views

Author

Paul D. Hanna, May 20 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 68*x^4 + 234*x^5 + 824*x^6 + 2956*x^7 + 10750*x^8 + 39540*x^9 + 146864*x^10 + ...

Links

Crossrefs

Cf. A363142, A363183, A363184, A363185.
Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (2*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).
(2) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 2*A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (2*A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 2*A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 2^(n-2*k) for n >= 0.
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