A359720 -id:A359720 - cp's OEIS frontend

A359722 a(n) = A359720(3n+1,2n) for n >= 0.

1, 9, 54, 269, 1254, 5642, 24828, 107613, 461318, 1961102, 8282196, 34792914, 145527004, 606473844, 2519619640, 10440010845, 43158028230, 178049440230, 733229991780, 3014712182790, 12377406450420, 50751988872780, 207859022097480, 850399040956530, 3475797671194524

Offset: 0

Paul D. Hanna, Jan 14 2023

nonn

The g.f. of A359720, G(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} A359720(n,k)*x^n*y^k, satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * G(x,y)^n.
A359720(3*n,2*n) = binomial(2*n+1,n)/(2*n+1) = A000108(n), n >= 0.
A359720(3*n+2,2*n+1) = binomial(2*n+1,n+1) + binomial(2*n+2,n), for n >= 0.

Cf. A359720, A000108.

/* a(n) = A359720(3*n+1,2*n) */
{a(n) = my(A=[1]); for(i=1, 3*n+1, A=concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^m * x^m * (y + x^m +x*O(x^#A) )^m * Ser(A)^m ), #A-1) );
polcoeff( polcoeff(Ser(A), 3*n+1,x), 2*n,y)}
for(n=0, 30, print1(a(n), ", "))

A355357 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)) A(x)^n.

Original entry on oeis.org

1, 1, 1, 4, 7, 20, 43, 110, 262, 674, 1684, 4397, 11320, 29938, 78641, 210044, 559724, 1507563, 4060585, 11016027, 29919220, 81673846, 223307300, 612851316, 1684816018, 4645243490, 12829177587, 35513736868, 98465916370, 273531234027, 760966444416

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

a(n) = Sum_{k=0..floor(n/2)} A355350(n-k,n-2*k) for n >= 0.

a(n) = A359720(n,0), for n >= 0.

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 20*x^5 + 43*x^6 + 110*x^7 + 262*x^8 + 674*x^9 + 1684*x^10 + 4397*x^11 + 11320*x^12 + ...
where
x = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 -+ ...
also,
x*P(x^2) = (1 - x^2*A(x))*(1 - 1/A(x)) * (1 - x^4*A(x))*(1 - x^2/A(x)) * (1 - x^6*A(x))*(1 - x^4/A(x)) * (1 - x^8*A(x))*(1 - x^6/A(x)) * ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355356.

Cf. A359720, A357221, A357222, A357223, A357224, A357225, A357226.

(* Calculation of constants {d,c}: *) {1/r, Sqrt[-Log[r] * ((-1 + r) * QPochhammer[1/r, r^2] * (-2*Log[r] + (-1 + r)*(Log[1 - r^2] - Log[r - r^3]) + (-1 + r) * QPolyGamma[0, -1/2, r^2] - (-1 + r)*QPolyGamma[0, 1, r^2]) + 4*(-1 + r)^2 * r^2 * Log[r] * Derivative[0, 1][QPochhammer][1/r, r^2] + 2*r^3 * Log[r] * QPochhammer[1/r, r^2]^3 * Derivative[0, 1][QPochhammer][r^2, r^2]) / (Pi*r^2* QPochhammer[1/r, r^2] * (-4*r*Log[r]^2 + (-1 + r)^2 * QPolyGamma[1, -1/2, r^2]))]} /. FindRoot[ 1/QPochhammer[r^2] == (r*QPochhammer[1/r, r^2]^2)/(-1 + r)^2, {r, 1/3}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Feb 01 2024 *)

{a(n) = my(A=[1,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(n+4));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) x*P(x^2) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
From Vaclav Kotesovec, Feb 01 2024: (Start)
Formula (2) can be rewritten as the functional equation x/QPochhammer(x^2) = QPochhammer(y, x^2)/(1 - y) * QPochhammer(1/(x^2*y), x^2)/(1 - 1/(x^2*y)).
a(n) ~ c * d^n / n^(3/2), where d = 2.92005174190265697439941308343193651904071627244119127019370275824199... and c = 1.4709989760845501303394202030872391136773745007487301056274536584990... (End)

A359670 Triangle of coefficients T(n,k) of x^ny^k in g.f. A(x,y) satisfying y = Sum_{n=-oo..+oo} (-1)^n x^n * (y*A(x,y) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 21, 12, 1, 14, 62, 68, 20, 1, 24, 162, 284, 170, 30, 1, 40, 384, 998, 970, 360, 42, 1, 64, 855, 3092, 4410, 2720, 679, 56, 1, 100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1, 154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1, 232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1

Offset: 0

Paul D. Hanna, Jan 17 2023

Related identity: 0 = Sum_{-oo..+oo} (-1)^n * x^n * (y + x^n)^n, which holds formally for all y.
T(n,0) = A015128(n), the number of overpartitions of n, for n >= 0.
T(n+1,1) = A022571(n), the coefficient of x^n in Product_{m>=1} (1 + x^m)^6, for n >= 0.
A359711(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A359712(n) = Sum_{k=0..n} T(n,k)*2^k for n >= 0.
A359713(n) = Sum_{k=0..n} T(n,k)*3^k for n >= 0.
A363104(n) = Sum_{k=0..n} T(n,k)*4^k for n >= 0.
A363105(n) = Sum_{k=0..n} T(n,k)*5^k for n >= 0.
A359714(n) = T(2*n,n) for n >= 0 (central terms).
A359715(n) = T(n+2,2) for n >= 0.
A359718(n) = T(n+3,3) for n >= 0.
A363142(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023
From Paul D. Hanna, May 20 2023: (Start)
A363182(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 2^(n-2*k) for n >= 0.
A363183(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 3^(n-2*k) for n >= 0.
A363184(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 4^(n-2*k) for n >= 0.
A363185(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 5^(n-2*k) for n >= 0. (End)

			G.f.: A(x,y) = 1 + x*(2 + y) + x^2*(4 + 6*y + y^2) + x^3*(8 + 21*y + 12*y^2 + y^3) + x^4*(14 + 62*y + 68*y^2 + 20*y^3 + y^4) + x^5*(24 + 162*y + 284*y^2 + 170*y^3 + 30*y^4 + y^5) + x^6*(40 + 384*y + 998*y^2 + 970*y^3 + 360*y^4 + 42*y^5 + y^6) + x^7*(64 + 855*y + 3092*y^2 + 4410*y^3 + 2720*y^4 + 679*y^5 + 56*y^6 + y^7) + x^8*(100 + 1806*y + 8724*y^2 + 17172*y^3 + 15627*y^4 + 6608*y^5 + 1176*y^6 + 72*y^7 + y^8) + x^9*(154 + 3648*y + 22904*y^2 + 59545*y^3 + 74682*y^4 + 47089*y^5 + 14392*y^6 + 1908*y^7 + 90*y^8 + y^9) + x^10*(232 + 7110*y + 56679*y^2 + 188700*y^3 + 311530*y^4 + 271698*y^5 + 125160*y^6 + 28764*y^7 + 2940*y^8 + 110*y^9 + y^10) + ...
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 0, k = 0..n, begins
[1];
[2, 1];
[4, 6, 1];
[8, 21, 12, 1];
[14, 62, 68, 20, 1];
[24, 162, 284, 170, 30, 1];
[40, 384, 998, 970, 360, 42, 1];
[64, 855, 3092, 4410, 2720, 679, 56, 1];
[100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1];
[154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1];
[232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1];
[344, 13434, 133516, 556085, 1169100, 1342684, 860664, 300888, 53640, 4345, 132, 1];
[504, 24702, 301664, 1542640, 4029237, 5884160, 4980320, 2438712, 666240, 94490, 6204, 156, 1];
[728, 44361, 657368, 4065868, 12940766, 23411339, 25215416, 16367874, 6302148, 1377464, 158708, 8606, 182, 1];
[1040, 78006, 1387854, 10253720, 39153924, 85994062, 114672768, 94919382, 48660900, 15071628, 2687454, 256022, 11648, 210, 1]; ...
RELATED SERIES.
Given g.f. F(x) of A361770, where
F(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ... + A361770(n)*x^n + ...
then
(1) F(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * F(x)^k,
(2) F(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^2 + x^(n-1))^(n+1).
Given g.f. G(x) of A363135, where
G(x) = 1 + 3*x + 17*x^2 + 133*x^3 + 1201*x^4 + 11796*x^5 + 122192*x^6 + 1314266*x^7 + 14536760*x^8 + 164299909*x^9 + ... + A363135(n)*x^n + ...
then
(1) G(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * G(x)^(2*k),
(2) G(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^3 + x^(n-1))^(n+1).

Paul D. Hanna, Table of n, a(n) for n = 0..2555

Cf. A359711 (row sums), A359712 (y=2), A359713 (y=3), A363104(y=4), A363105 (y=5).
Cf. A359714 (central terms), A359715 (column 2), A359718 (column 3).
Cf. A363142, A363182, A363183, A363184, A363185.
Cf. A361770, A363135, A363136, A363137.
Cf. A359720, A293600.

{T(n,k) = my(A=1); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( polcoeff( A,n,x),k,y)}
for(n=0,15, for(k=0,n, print1( T(n,k),", "));print(""))

{T(n,k) = my(A=[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) ); polcoeff( A[n+1],k,y)}
for(n=0,15, for(k=0,n, print1( T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k may be described as follows.
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).
(2) x*y = Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^(n+1).
(3) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n-1).
(4) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^n ].
(5) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^(n+1))^n ].
From Paul D. Hanna, May 18 2023: (Start)
(6) y = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (y*A(x,y) + x^n)^n.
(7) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (y*A(x,y) + x^n)^n ].
(8) x*y = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (y*A(x,y) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^n. (End)

A357797 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 5, 18, 85, 374, 1659, 7774, 36876, 177494, 867424, 4285653, 21373782, 107475746, 544244911, 2773091748, 14207171278, 73140904609, 378184133959, 1963127909395, 10226682384980, 53446907352828, 280150058149086, 1472424136948438, 7758105323877698, 40970959715619200, 216830651728330127

Offset: 0

Paul D. Hanna, Dec 22 2022

nonn

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

			G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 85*x^4 + 374*x^5 + 1659*x^6 + 7774*x^7 + 36876*x^8 + 177494*x^9 + 867424*x^10 + 4285653*x^11 + 21373782*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A359720, A359721, A359723, A357798.

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

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 2*x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*2^k, for n >= 0.

A359721 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 3, 10, 37, 127, 460, 1710, 6461, 24851, 96921, 382358, 1522997, 6116518, 24740564, 100698617, 412126133, 1694982357, 7001729420, 29037602898, 120856092153, 504647152650, 2113469775619, 8875358529059, 37364827472930, 157668052571948, 666735804080597, 2825054673048981

Offset: 0

Paul D. Hanna, Jan 11 2023

nonn

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 127*x^5 + 460*x^6 + 1710*x^7 + 6461*x^8 + 24851*x^9 + 96921*x^10 + 382358*x^11 + 1522997*x^12 + ...
SPECIFIC VALUES.
A(2/9) = 2.24070435506724977359903344036738515875266644317987374...
A(x) = 2 at x = 0.21791735938682393028374635435485389216073583164032813...
A(1/5) = 1.63325728843716074555468074513852677972333543319428229...
A(1/6) = 1.36828627213340815002770404510072582545059876619425902...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A359720, A355357, A357797, A359723, A359724.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k), for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 4.470597712126170109... and c = 1.18164918660560739... - Vaclav Kotesovec, Mar 14 2023

A359723 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 7, 28, 151, 803, 4108, 22532, 125449, 705929, 4035955, 23332364, 136111591, 800561116, 4741777880, 28258286033, 169322163149, 1019483819757, 6164900341534, 37425357962592, 228002416106605, 1393503512669230, 8541839907812651, 52500559705299795, 323483846045526418

Offset: 0

Paul D. Hanna, Jan 11 2023

nonn

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

			G.f.: A(x) = 1 + x + 7*x^2 + 28*x^3 + 151*x^4 + 803*x^5 + 4108*x^6 + 22532*x^7 + 125449*x^8 + 705929*x^9 + 4035955*x^10 + 23332364*x^11 + 136111591*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.150684304746792807618050217238804920801612774142866...
A(1/7) = 1.67848119643298635311797131334138331526984303696733717...
A(1/8) = 1.40389487408504106142147713148599989460789630965507028...

Cf. A359720, A355357, A357797, A359721, A359724.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 3*x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*3^k, for n >= 0.

A359724 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 9, 40, 235, 1456, 8323, 51510, 324674, 2061746, 13308492, 86876405, 572169044, 3799139674, 25403610485, 170901457100, 1155976005944, 7856772779823, 53630378512469, 367507023955203, 2527254094342404, 17435029150904202, 120633291776867632, 836907189915348056

Offset: 0

Paul D. Hanna, Jan 11 2023

nonn

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

			G.f.: A(x) = 1 + x + 9*x^2 + 40*x^3 + 235*x^4 + 1456*x^5 + 8323*x^6 + 51510*x^7 + 324674*x^8 + 2061746*x^9 + 13308492*x^10 + 86876405*x^11 + 572169044*x^12 + ...

Cf. A359720, A355357, A357797, A359721, A359723.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 4*x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*4^k, for n >= 0.

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A359725 a(n) = A359720(n+2,1), for n >= 0.

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A359726 a(n) = A359720(n+3,2), for n >= 0.

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A359722 a(n) = A359720(3*n+1,2*n) for n >= 0.

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A355357 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

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A359670 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) satisfying y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).

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A357797 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.

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A359721 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.

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A359723 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.

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A359722 a(n) = A359720(3n+1,2n) for n >= 0.

A355357 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)) A(x)^n.

A359670 Triangle of coefficients T(n,k) of x^ny^k in g.f. A(x,y) satisfying y = Sum_{n=-oo..+oo} (-1)^n x^n * (y*A(x,y) + x^(n-1))^(n+1).