A355350 -id:A355350 - cp's OEIS frontend

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

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)

A355351 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 1, 4, 16, 60, 231, 920, 3819, 16365, 71792, 320219, 1446517, 6602975, 30415725, 141231704, 660431602, 3107519738, 14701758926, 69891556656, 333700223891, 1599475107712, 7693580712200, 37125486197570, 179675330190428, 871910824853956, 4241603521253775

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

a(n) = Sum_{k=0..n} A355350(n,k) for n >= 0.

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + 16365*x^8 + 71792*x^9 + 320219*x^10 + 1446517*x^11 + ...
where
x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/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..600

Cf. A355350, A355352, A355353, A355354, A355355, A355356, A355357.

(* Calculation of constants {d,c}: *) {1/r, s*Sqrt[(-1 + s)*(-1 + r*s) * Log[r]* ((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r] - r*(-1 + s)*(-1 + r*s)*Log[r]* Derivative[0, 1][QPochhammer][r, r] / QPochhammer[r] + r*s*Log[r] * QPochhammer[r] * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + (-1 + r*s) * ((1 - s) * QPolyGamma[0, Log[s]/Log[r], r] + Log[r] * (-1 - (r*(-1 + s) * Derivative[0, 1][QPochhammer][s, r]) / QPochhammer[s, r]))) / (-s*(1 + r - 4*r*s + r*(1 + r)*s^2) * Log[r]^2 + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, -Log[r*s]/Log[r], r]) / (2*Pi)]} /. FindRoot[{1/QPochhammer[r] + s*QPochhammer[1/(r*s), r] * QPochhammer[s, r] / ((-1 + s) * (-1 + r*s)) == 0, (-1 + r*s^2)*Log[r] + (-1 + s) * (-1 + r*s) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s] / Log[r], r]) == 0}, {r, 1/5}, {s, 2}, 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(2*n+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*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)/2) * A(x)^n.
(2) x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/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) = QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)).
a(n) ~ c * d^n / n^(3/2), where d = 5.163920888936085556632234304058129... and c = 0.824708825453794494929019119272... (End)

A355352 G.f. A(x) satisfies: 2x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 2, 10, 50, 248, 1294, 7092, 40426, 236698, 1412860, 8561906, 52546920, 326011118, 2041512624, 12886608654, 81908498582, 523780469070, 3367399778356, 21752611767804, 141118852010146, 919035717462824, 6006146649948722, 39376700396145616, 258907024677687808

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

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

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 248*x^4 + 1294*x^5 + 7092*x^6 + 40426*x^7 + 236698*x^8 + 1412860*x^9 + 8561906*x^10 + ...
where
2*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
2*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/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, A355353, A355354, A355355, A355356, A355357.

{a(n) = my(A=[1,2],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 2*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355353 G.f. A(x) satisfies: 3x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 3, 18, 108, 660, 4275, 29106, 205377, 1485279, 10943424, 81866493, 620316297, 4751289063, 36727782675, 286153810542, 2244799306134, 17715992048886, 140560480602810, 1120518766292436, 8970573523101477, 72091628161825608, 581375787259765554, 4703286596619094686

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

a(n) = Sum_{k=0..n} A355350(n,k) * 3^k for n >= 0.

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 108*x^3 + 660*x^4 + 4275*x^5 + 29106*x^6 + 205377*x^7 + 1485279*x^8 + 10943424*x^9 + 81866493*x^10 + ...
where
3*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
3*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/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, A355354, A355355, A355356, A355357.

{a(n) = my(A=[1,3],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 3*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355354 G.f. A(x) satisfies: 4x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 4, 28, 196, 1416, 10860, 87392, 727188, 6196212, 53783336, 474011756, 4231158016, 38174676188, 347566170384, 3189295781780, 29465038957708, 273851282010308, 2558703740102840, 24019990008557160, 226444571054525156, 2142925363606256584, 20349477565111498148

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

a(n) = Sum_{k=0..n} A355350(n,k) * 4^k for n >= 0.

			G.f.: A(x) = 1 + 4*x + 28*x^2 + 196*x^3 + 1416*x^4 + 10860*x^5 + 87392*x^6 + 727188*x^7 + 6196212*x^8 + 53783336*x^9 + 474011756*x^10 + ...
where
4*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
4*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/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, A355355, A355356, A355357.

{a(n) = my(A=[1,4],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 4*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355355 G.f. A(x) satisfies: 5x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 5, 40, 320, 2660, 23455, 216540, 2064055, 20137945, 200134600, 2019406895, 20635313325, 213109960895, 2220820915065, 23323755734820, 246616999661690, 2623193780773530, 28049464032800110, 301340494687086960, 3251017466141039095, 35207152686408604400

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

a(n) = Sum_{k=0..n} A355350(n,k) * 5^k for n >= 0.

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 320*x^3 + 2660*x^4 + 23455*x^5 + 216540*x^6 + 2064055*x^7 + 20137945*x^8 + 200134600*x^9 + 2019406895*x^10 + ...
where
5*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
5*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/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, A355356, A355357.

{a(n) = my(A=[1,5],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) 5*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22, 0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140, 0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969, 0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084, 0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820, 0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732

Offset: 0

Paul D. Hanna, Aug 09 2022

T(n, 3*n+1) = [x^n*y^(3*n+1)] A(x,y) = binomial(4*n, n)/(3*n + 1) = A002293(n) for n >= 0, where g.f. G(x) of A002293 satisfies: G(x) = 1 + x*G(x)^4.
T(4*n, 1) = A000716(n) for n >= 0 (nonzero terms in column 1).
T(4*n+3, 2) = [x^(4*n+3)*y^2] A(x,y) = 2 * A354655(n+1) for n >= 0, where A354655 equals column 2 of triangle A354650.
T(4*n+2, 3) = [x^(4*n+2)*y^3] A(x,y) = 4 * A354656(n+1) for n >= 0, where A354656 equals column 3 of triangle A354650.
T(2*n, 2*n+1) = A356504(n), for n >= 0.
T(2*n+1, 2*n) = A356505(n) for n >= 0.
T(3*n, n+1) = A356506(n) for n >= 0.
T(3*n+1, n) = A355365(n) where A355365 is the central terms of A355360 = A355360(2*n,n).
Sum_{k=0..3*n+1} T(n, k) = A354248(n) for n >= 0 (row sums).
Sum_{k=0..3*n+1} T(n, k) * 2^k = A356502(n) for n >= 0.
Sum_{k=0..3*n+1} T(n, k) * 3^k = A356503(n) for n >= 0.
Sum_{k=0..3*n+1} T(4*n+1-k, k) = A355872(n+1) for n >= 0 (nonzero antidiagonal sums).
SPECIFIC VALUES.
(V.1) 1 = A(x,y) at x = -exp(-Pi) and y = Pi^(1/4)/gamma(3/4).
(V.2) 1 = A(x,y) at x = -exp(-2*Pi) and y = Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) 1 = A(x,y) at x = -exp(-3*Pi) and y = Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) 1 = A(x,y) at x = -exp(-4*Pi) and y = Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) 1 = A(x,y) at x = -exp(-sqrt(3)*Pi) and y = gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = y + x*(1 + y^4) + x^2*(4*y^3 + 4*y^7) + x^3*(6*y^2 + 28*y^6 + 22*y^10) + x^4*(3*y + 84*y^5 + 219*y^9 + 140*y^13) + x^5*(135*y^4 + 981*y^8 + 1807*y^12 + 969*y^16) + x^6*(120*y^3 + 2568*y^7 + 10764*y^11 + 15368*y^15 + 7084*y^19) + x^7*(54*y^2 + 4284*y^6 + 38896*y^10 + 114240*y^14 + 133266*y^18 + 53820*y^22) + x^8*(9*y + 4662*y^5 + 94390*y^9 + 525980*y^13 + 1187433*y^17 + 1171390*y^21 + 420732*y^25) + x^9*(3250*y^4 + 160965*y^8 + 1670942*y^12 + 6640711*y^16 + 12167001*y^20 + 10399545*y^24 + 3362260*y^28) + ...
such that A = A(x,y) satisfies
y = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
This irregular triangle of coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
  n = 0: [0, 1];
  n = 1: [1, 0, 0, 0, 1];
  n = 2: [0, 0, 0, 4, 0, 0, 0, 4];
  n = 3: [0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22];
  n = 4: [0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140];
  n = 5: [0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969];
  n = 6: [0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084];
  n = 7: [0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820];
  n = 8: [0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732];
  n = 9: [0, 0, 0, 0, 3250, 0, 0, 0, 160965, 0, 0, 0, 1670942, 0, 0, 0, 6640711, 0, 0, 0, 12167001, 0, 0, 0, 10399545, 0, 0, 0, 3362260];
  ...
Reading this triangle by nonzero antidiagonals [x^(4*n+1-k)*y^k] A(x,y) for n >= 0, k = 0..3*n+1, yields triangle A356501:
  [1, 1];
  [0, 3, 6, 4, 1];
  [0, 9, 54, 120, 135, 84, 28, 4];
  [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
  [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
  [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..3926

Cf. A354248, A356502, A356503, A356504, A356505, A355872.
Cf. A354655, A354656, A355365, A000716, A002293.
Cf. related tables: A356501, A355350, A355360, A355870.

{T(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,9, for(k=0,3*n+1, print1(T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).
(2) y = A(x,y) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n-1)*A(x,y)^(2*n+1)) * (1 - x^(2*n-1)*A(x,y)^(2*n-3)), by the Jacobi triple product identity.
(3) y = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n+1)*A(x,y)^(2*n-1)) * (1 - x^(2*n-3)*A(x,y)^(2*n-1)), by the Jacobi triple product identity.
(4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).
(5) 1 = A(x, theta_4(x)) where theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2) is a Jacobi theta function.

A355356 G.f. A(x) satisfies: x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 0, 1, 3, 10, 28, 79, 216, 603, 1702, 4933, 14620, 44287, 136352, 424858, 1334162, 4211572, 13344072, 42412667, 135217722, 432483522, 1387929369, 4469341807, 14439523193, 46795072968, 152076428228, 495460089510, 1617787324674, 5292984017236, 17348743335252

Offset: 0

Paul D. Hanna, Jun 29 2022

nonn

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

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 10*x^4 + 28*x^5 + 79*x^6 + 216*x^7 + 603*x^8 + 1702*x^9 + 4933*x^10 + 14620*x^11 + 44287*x^12 + ...
where
x^2 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
x^2*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/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 + ...

Cf. A355350, A355351, A355352, A355353, A355354, A355355, A355357.

{a(n) = my(A=[1,0,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x^2 - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) x^2*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A355360 G.f. A(x,y) satisfies: xyA(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 9, 12, 5, 0, 22, 54, 46, 14, 0, 51, 196, 282, 175, 42, 0, 108, 630, 1360, 1365, 666, 132, 0, 221, 1836, 5635, 8190, 6321, 2541, 429, 0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430, 0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862, 0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796

Offset: 0

Paul D. Hanna, Jul 19 2022

The main diagonal equals A000108, the Catalan numbers.
Conjectures.
(C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds.
(C.2) Column 2 equals twice A023005, the number of partitions into parts of 6 kinds.
The term T(n,k) is found in row n and column k of this triangle, and can be used to derive the following sequences.
A355361(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A355362(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A355363(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A355364(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).
A355365(n) = T(2*n,n) for n >= 0 (central terms of this triangle).

			G.f.: A(x,y) = 1 + x*y + x^2*(3*y + 2*y^2) + x^3*(9*y + 12*y^2 + 5*y^3) + x^4*(22*y + 54*y^2 + 46*y^3 + 14*y^4) + x^5*(51*y + 196*y^2 + 282*y^3 + 175*y^4 + 42*y^5) + x^6*(108*y + 630*y^2 + 1360*y^3 + 1365*y^4 + 666*y^5 + 132*y^6) + x^7*(221*y + 1836*y^2 + 5635*y^3 + 8190*y^4 + 6321*y^5 + 2541*y^6 + 429*y^7) + x^8*(429*y + 4984*y^2 + 20850*y^3 + 41405*y^4 + 45326*y^5 + 28448*y^6 + 9724*y^7 + 1430*y^8) + x^9*(810*y + 12744*y^2 + 70737*y^3 + 184527*y^4 + 270060*y^5 + 237209*y^6 + 125532*y^7 + 37323*y^8 + 4862*y^9) + x^10*(1479*y + 31050*y^2 + 223652*y^3 + 745745*y^4 + 1404102*y^5 + 1625932*y^6 + 1193116*y^7 + 546039*y^8 + 143650*y^9 + 16796*y^10) + ...
where
x*y*A(x) = ... - x^10/A(x,y)^5 + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
x*y*A(x)*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ...
TRIANGLE.
The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins:
n=0: [1];
n=1: [0, 1];
n=2: [0, 3, 2];
n=3: [0, 9, 12, 5];
n=4: [0, 22, 54, 46, 14];
n=5: [0, 51, 196, 282, 175, 42];
n=6: [0, 108, 630, 1360, 1365, 666, 132];
n=7: [0, 221, 1836, 5635, 8190, 6321, 2541, 429];
n=8: [0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430];
n=9: [0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862];
n=10: [0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796];
n=11: [0, 2640, 72560, 667005, 2784110, 6565030, 9646462, 9242178, 5826171, 2349490, 554268, 58786];
n=12: [0, 4599, 163632, 1892670, 9729720, 28161819, 51126740, 61555824, 50308245, 27806065, 10023948, 2143428, 208012];
...
in which column 1 appears to equal A000716, the coefficients in P(x)^3,
and column 2 appears to equal twice A023005, the coefficients in P(x)^6,
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 + ... + A000041(n)*x^n + ...
Also, the power series expansions of P(x)^3 and P(x)^6 begin
P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ...
P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ...
The main diagonal equals the Catalan numbers (A000108), where g.f. C(x) = 1 + x*C(x)^2 begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..5150

Cf. A000108 (main diagonal), A000041, A000716, A023005.
Cf. A355361 (y=1), A355362 (y=2), A355363 (y=3), A355364, A355365.
Cf. A355350 (related table).

{T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( x*y*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:
(1) x*y*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) -x*y*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x,y)^n.
(3) x*y*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
(4) A(x,y) = B(x, y*A(x,y)) and A(x, y/B(x,y)) = B(x,y) where x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * B(x,y)^n, and B(x,y) is the g.f. of table A355350.

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2n+1) Sum_{k=0..3n} T(n,k)y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n.

Original entry on oeis.org

1, 0, 3, -3, 1, 0, 9, -18, 21, -15, 6, -1, 0, 22, -56, 116, -182, 196, -140, 64, -17, 2, 0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5, 0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14, 0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42

Offset: 0

Paul D. Hanna, Jul 19 2022

Row sums equal A000108, the Catalan numbers:
Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.
T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).
Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).
The generating functions of some related sequences are given as follows.
(1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.
(2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.
(3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.
(4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.
(5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.
(6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.
(7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.
(8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.
(9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.
(10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.
SPECIFIC VALUES.
(V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).
(V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...
where
y = ... + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...
This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins
n = 0: [1];
n = 1: [0, 3, -3, 1];
n = 2: [0, 9, -18, 21, -15, 6, -1];
n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];
n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];
n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];
n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];
n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...
The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.
Column 1 appears to equal A000716 (ignoring the initial term).
Example: at y = x, we have the g.f. of A355351:
A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...
where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

Paul D. Hanna, Table of n, a(n) for n = 0..3875

Cf. A000108 (row sums), A355871 (y=2).
Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).
Cf. A355360 (related triangle), A000716.

{T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}
for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

cp's OEIS Frontend

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

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A355351 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355352 G.f. A(x) satisfies: 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355353 G.f. A(x) satisfies: 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355354 G.f. A(x) satisfies: 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355355 G.f. A(x) satisfies: 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A356500 Coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

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

A355351 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

A355352 G.f. A(x) satisfies: 2x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A355353 G.f. A(x) satisfies: 3x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A355354 G.f. A(x) satisfies: 4x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A355355 G.f. A(x) satisfies: 5x = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

A355356 G.f. A(x) satisfies: x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

A355360 G.f. A(x,y) satisfies: xyA(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2n+1) Sum_{k=0..3n} T(n,k)y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n.