A354650 -id:A354650 - cp's OEIS frontend

A354655 Column 2 of triangle A354650: a(n) = A354650(n,2), for n >= 1.

3, 27, 147, 630, 2295, 7476, 22302, 62100, 163260, 409080, 983367, 2280306, 5122026, 11184075, 23806575, 49521456, 100872135, 201558231, 395675475, 764130780, 1453424259, 2725614243, 5044092372, 9219499800, 16655554125, 29759775435, 52623867051

Offset: 1

Paul D. Hanna, Jun 02 2022

nonn

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A354649, A354650, A354656.

{A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=1,30,print1(A354650(n,2),", "))

a(n) = (-1)^(n+1) * A354649(n,2), for n >= 1.

a(n) = A354650(n,2), for n >= 1.

A354658 A diagonal of triangle A354650: a(n) = A354650(n,n), for n >= 0.

Original entry on oeis.org

1, 3, 27, 340, 5070, 83559, 1472261, 27205308, 520974180, 10257025240, 206469879462, 4232227325352, 88073315164471, 1856404180514940, 39560345751767970, 851083806077023888, 18462636758298743712, 403459312929849694791, 8874351725505564788350

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A354649, A354650, A354659, A354660.

{A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,n),", "))

a(n) = -A354649(n,n), for n >= 0.
a(n) = A354650(n,n), for n >= 0.
a(n) ~ c * d^n / n^2, where d = 24.575992877869992813144975... and c = 0.285171824264368179079895... - Vaclav Kotesovec, Jun 08 2022

A354656 Column 3 of triangle A354650: a(n) = A354650(n,3), for n >= 1.

Original entry on oeis.org

1, 30, 340, 2530, 14595, 70737, 301070, 1157820, 4100785, 13563010, 42321840, 125586440, 356621070, 973989030, 2569116330, 6567458520, 16317741975, 39504992395, 93390535840, 215983566780, 489454806785, 1088433416785, 2378160809610, 5111208572940, 10816601842950

Offset: 1

Paul D. Hanna, Jun 02 2022

nonn

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A354649, A354650, A354655, A354657.

{A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=1,30,print1(A354650(n,3),", "))

a(n) = (-1)^n * A354649(n,3), for n >= 1.

a(n) = A354650(n,3), for n >= 1.

A354659 A diagonal of triangle A354650: a(n) = A354650(n,n+1), for n >= 0.

Original entry on oeis.org

1, 3, 30, 390, 5928, 98910, 1757688, 32683680, 628884300, 12428334215, 250940544738, 5156722096422, 107538413657010, 2270751678647100, 48464836803383400, 1044050265679857144, 22675350105240015204, 496034970650911331550, 10920742396832034391590

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A354649, A354650, A354658, A354660.

{A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,n+1),", "))

a(n) = A354649(n,n+1), for n >= 0.
a(n) = A354650(n,n+1), for n >= 0.
a(n) ~ c * d^n / n^2, where d = 24.5759928778699928131449756... and c = 0.35661791857107638456206... - Vaclav Kotesovec, Mar 19 2023

A354660 a(n) = A354650(n,2*n), for n >= 0.

Original entry on oeis.org

1, 3, 15, 83, 486, 2937, 18109, 113220, 715122, 4552229, 29156985, 187683795, 1213110600, 7868238588, 51184173036, 333809308696, 2181842704602, 14288748463485, 93737673347185, 615889045662345, 4052198020223430, 26694405836621985, 176052003674681925

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A354649, A354650, A354658, A354659.

{A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,2*n),", "))

a(n) = (-1)^(n+1) * A354649(n,2*n), for n >= 0.
a(n) = A354650(n,2*n), for n >= 0.
a(n) ~ 13 * 3^(3*n - 5/2) / (sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Mar 19 2023

A354649 G.f. A(x,y) satisfies: y = f(x,A(x,y)), where f(x,y) = Sum_{n=-oo..oo} x^(n(n+1)/2) y^(n*(n-1)/2) is Ramanujan's theta function.

Original entry on oeis.org

-1, 1, 0, -3, 3, -1, 0, 9, -27, 30, -15, 3, 0, -22, 147, -340, 390, -246, 83, -12, 0, 51, -630, 2530, -5070, 5928, -4284, 1908, -486, 55, 0, -108, 2295, -14595, 45450, -83559, 98910, -78282, 41580, -14355, 2937, -273, 0, 221, -7476, 70737, -319605, 849450, -1472261, 1757688, -1484451, 891890, -375442, 105930, -18109, 1428, 0, -429, 22302, -301070, 1886010, -6878907, 16386636, -27205308, 32683680, -28981855, 19081854, -9258678, 3231514, -771225, 113220, -7752

Offset: 0

Paul D. Hanna, Jun 02 2022

Signed version of A354650.
Column 1 equals signed A000716, with g.f. P(-x)^3 where P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ) is the partition function.
The rightmost border equals signed A001764, with g.f. C(x) = 1 - x*C(x)^3.
T(n,1) = (-1)^n * A000716(n), for n >= 0.
T(n,2) = (-1)^(n+1) * A354655(n), for n >= 1.
T(n,3) = (-1)^n * A354656(n), for n >= 1.
T(n,n) = -A354658(n), for n >= 0.
T(n,n+1) = A354659(n), for n >= 0.
T(n,2*n) = (-1)^(n+1) * A354660(n), for n >= 0.
T(n,2*n+1) = (-1)^n * A001764(n), for n >= 0.
Antidiagonal sums equals signed A268650.
Sum_{k=0..2*n+1} T(n,k)*(-1)^k = (-1)^(n+1) * A268299(n+1), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-2)^k = (-1)^(n+1) * A354652(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-3)^k = (-1)^(n+1) * A354653(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-4)^k = (-1)^(n+1) * A354654(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k) = (-1)^n * A354661(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*2^k = (-1)^n * A354662(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*3^k = (-1)^n * A354663(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*4^k = (-1)^n * A354664(n), for n >= 0.
SPECIFIC VALUES.
(1) A(x,y) = exp(-Pi) at x = exp(-Pi), y = Pi^(1/4)/gamma(3/4).
(2) A(x,y) = exp(-2*Pi) at x = exp(-2*Pi), y = Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(3) A(x,y) = exp(-3*Pi) at x = exp(-3*Pi), y = Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(4) A(x,y) = exp(-4*Pi) at x = exp(-4*Pi), y = Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(5) A(x,y) = exp(-sqrt(3)*Pi) at x = exp(-sqrt(3)*Pi), y = gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = (-1 + y) - x*(3*y - 3*y^2 + y^3) + x^2*(9*y - 27*y^2 + 30*y^3 - 15*y^4 + 3*y^5) - x^3*(22*y - 147*y^2 + 340*y^3 - 390*y^4 + 246*y^5 - 83*y^6 + 12*y^7) + x^4*(51*y - 630*y^2 + 2530*y^3 - 5070*y^4 + 5928*y^5 - 4284*y^6 + 1908*y^7 - 486*y^8 + 55*y^9) - x^5*(108*y - 2295*y^2 + 14595*y^3 - 45450*y^4 + 83559*y^5 - 98910*y^6 + 78282*y^7 - 41580*y^8 + 14355*y^9 - 2937*y^10 + 273*y^11) + x^6*(221*y - 7476*y^2 + 70737*y^3 - 319605*y^4 + 849450*y^5 - 1472261*y^6 + 1757688*y^7 - 1484451*y^8 + 891890*y^9 - 375442*y^10 + 105930*y^11 - 18109*y^12 + 1428*y^13) + x^7*(-429*y + 22302*y^2 - 301070*y^3 + 1886010*y^4 - 6878907*y^5 + 16386636*y^6 - 27205308*y^7 + 32683680*y^8 - 28981855*y^9 + 19081854*y^10 - 9258678*y^11 + 3231514*y^12 - 771225*y^13 + 113220*y^14 - 7752*y^15) + x^8*(810*y - 62100*y^2 + 1157820*y^3 - 9729720*y^4 + 46977378*y^5 - 147584556*y^6 + 324283068*y^7 - 520974180*y^8 + 628884300*y^9 - 579226362*y^10 + 409367712*y^11 - 221218179*y^12 + 90115620*y^13 - 26879160*y^14 + 5559408*y^15 - 715122*y^16 + 43263*y^17) + ...
such that A = A(x,y) satisfies:
(1) y = ... + x^36*A^28 + x^28*A^21 + x^21*A^15 + x^15*A^10 + x^10*A^6 + x^6*A^3 + x^3*A + x + 1 + A + x*A^3 + x^3*A^6 + x^6*A^10 + x^10*A^15 + x^15*A^21 + x^21*A^28 + x^28*A^36 + ...
(2) y = (1 - x*A)*(1 + A)*(1+x) * (1 - x^2*A^2)*(1 + x*A^2)*(1 + x^2*A) * (1 - x^3*A^3)*(1 + x^2*A^3)*(1 + x^3*A^2) * (1 - x^4*A^4)*(1 + x^3*A^4)*(1 + x^4*A^3) * (1 - x^5*A^5)*(1 + x^4*A^5)*(1 + x^5*A^4) * ...
(3) y = (1+x) + (1+x^3)*A + x*(1+x^5)*A^3 + x^3*(1+x^7)*A^6 + x^6*(1+x^9)*A^10 + x^10*(1+x^11)*A^15 + x^15*(1+x^13)*A^21 + x^21*(1+x^15)*A^28 + ...
(4) y = (1+A) + (1+A^3)*x + A*(1+A^5)*x^3 + A^3*(1+A^7)*x^6 + A^6*(1+A^9)*x^10 + A^10*(1+A^11)*x^15 + A^15*(1+A^13)*x^21 + A^21*(1+A^15)*x^28 + ...
This triangle of coefficients of x^n*y^k in g.f. A(x,y) for n >= 0, k = 0..2*n+1, begins:
-1, 1;
0, -3, 3, -1;
0, 9, -27, 30, -15, 3;
0, -22, 147, -340, 390, -246, 83, -12;
0, 51, -630, 2530, -5070, 5928, -4284, 1908, -486, 55;
0, -108, 2295, -14595, 45450, -83559, 98910, -78282, 41580, -14355, 2937, -273;
0, 221, -7476, 70737, -319605, 849450, -1472261, 1757688, -1484451, 891890, -375442, 105930, -18109, 1428;
0, -429, 22302, -301070, 1886010, -6878907, 16386636, -27205308, 32683680, -28981855, 19081854, -9258678, 3231514, -771225, 113220, -7752;
0, 810, -62100, 1157820, -9729720, 46977378, -147584556, 324283068, -520974180, 628884300, -579226362, 409367712, -221218179, 90115620, -26879160, 5559408, -715122, 43263; ...
The rightmost border equals signed A001764, with g.f. C(x) = 1 - x*C(x)^3.
Column 1 equals signed A000716, with g.f. P(-x)^3 where P(x) = exp( Sum_{n>=1} x^n/(n*(1-x^n)) ) is the partition function.

Paul D. Hanna, Table of n, a(n) for n = 0..10301

Cf. A000716 (column 1), A354655 (column 2), A354656 (column 3).
Cf. A354658 (T(n,n)), A354659 (T(n,n+1)), A354660 (T(n,2*n)), A001764 (right border).
Cf. A268299 (y=-1), A354652 (y=-2), A354653 (y=-3), A354654 (y=-4).
Cf. A354661 (y=1), A354662 (y=2), A354663 (y=3), A354664 (y=4).
Cf. A268650 (antidiagonal sums), A354657, A354650.

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

G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..2*n+1} T(n,k)*y^k satisfies:
(1) y = A(x,f(x,y)) = Sum_{n>=0} x^n * Sum_{k=0..2*n+1} T(n,k) * f(x,y)^k, where f(,) is Ramanujan's theta function.
(2) y = f(x,A(x,y)) = Sum_{n=-oo..oo} x^(n*(n-1)/2) * A(x,y)^(n*(n+1)/2), where f(,) is Ramanujan's theta function.
(3) y = Product_{n>=1} (1 - x^n*A(x,y)^n) * (1 + x^(n-1)*A(x,y)^n) * (1 + x^n*A(x,y)^(n-1)), by the Jacobi triple product identity.
(4) y = Sum_{n>=0} x^(n*(n-1)/2) * (1 + x^(2*n+1)) * A(x,y)^(n*(n+1)/2).
(5) y = Sum_{n>=0} A(x,y)^(n*(n-1)/2) * (1 + A(x,y)^(2*n+1)) * x^(n*(n+1)/2).
(6) T(n,1) = (-1)^n * A000716(n), where A000716(n) is the number of partitions of n into parts of 3 kinds.
(7) T(n,2*n+1) = (-1)^n * A001764(n) = (-1)^n * binomial(3*n,n)/(2*n+1), for n >= 0.

A268299 G.f. A(x) satisfies: -1 = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n/x) * (1 - A(x)^(n-1)*x).

Original entry on oeis.org

2, 7, 84, 1240, 20942, 382344, 7354688, 146810440, 3012778758, 63167322872, 1347251937632, 29138746861200, 637584335364362, 14088532800477752, 313936020646727040, 7046500093908958288, 159171390375064583380, 3615669944253537267048, 82541551931101193203004, 1892725670848222011475776, 43575217427267416453289838, 1006843304895182755611475824, 23340548167572913996786290328

Offset: 1

Paul D. Hanna, Feb 26 2016

nonn

The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

Equals the row sums of triangle A354650. - Paul D. Hanna, Jul 27 2022

			G.f.: A(x) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...
where A(x) satisfies the Jacobi Triple Product:
-1 = (1-A(x))*(1-A(x)/x)*(1-x) * (1-A(x)^2)*(1-A(x)^2/x)*(1-A(x)*x) * (1-A(x)^3)*(1-A(x)^3/x)*(1-A(x)^2*x) * (1-A(x)^4)*(1-A(x)^4/x)*(1-A(x)^3*x) * (1-A(x)^5)*(1-A(x)^5/x)*(1-A(x)^4*x) * (1-A(x)^6)*(1-A(x)^6/x)*(1-A(x)^5*x) +...
also
1/x = (1-A(x))*(1-A(x)*x)*(1-1/x) * (1-A(x)^2)*(1-A(x)^2*x)*(1-A(x)/x) * (1-A(x)^3)*(1-A(x)^3*x)*(1-A(x)^2/x) * (1-A(x)^4)*(1-A(x)^4*x)*(1-A(x)^3/x) * (1-A(x)^5)*(1-A(x)^5*x)*(1-A(x)^4/x) * (1-A(x)^6)*(1-A(x)^6*x)*(1-A(x)^5/x) *...
further,
-1 = (1-x) - A(x)*(1-x^3)/x + A(x)^3*(1-x^5)/x^2 - A(x)^6*(1-x^7)/x^3 + A(x)^10*(1-x^9)/x^4 - A(x)^15*(1-x^11)/x^5 + A(x)^21*(1-x^13)/x^6 +...
RELATED SERIES.
The series reversion of g.f. A(x) equals x*Q(x), where Q(x) begins:
Q(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 - 1979248977748/2*x^10/4^10 +...+ A268301(n)/2*x^n/4^n +...
and where Q(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x*Q(x))*(1-1/Q(x)) * (1-x^2)*(1-x^2*Q(x))*(1-x/Q(x)) * (1-x^3)*(1-x^3*Q(x))*(1-x^2/Q(x)) * (1-x^4)*(1-x^4*Q(x))*(1-x^3/Q(x)) * (1-x^5)*(1-x^5*Q(x))*(1-x^4/Q(x)) * (1-x^6)*(1-x^6*Q(x))*(1-x^5/Q(x)) *...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A354650, A268300, A268301, A268650.

(* Calculation of constant d: *) 1/r /. FindRoot[{r*QPochhammer[1/r, s]*QPochhammer[r, s]* QPochhammer[s, s] == 1 - r, (Log[1 - s] + QPolyGamma[0, 1, s])/(s*Log[s]) - Derivative[0, 1][QPochhammer][1/r, s]/QPochhammer[1/r, s] - Derivative[0, 1][QPochhammer][r, s]/QPochhammer[r, s] - Derivative[0, 1][QPochhammer][s, s]/ QPochhammer[s, s] == 0}, {r, 1/24}, {s, 1/8}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)

{a(n) = my(Q=1/2, t=floor(sqrt(2*n+1)+1/2)); for(i=0, n, Q = (Q + sum(m=-t, t, x^(m*(m-1)/2) * (-Q)^m +x*O(x^n)) )/2 ); polcoeff(serreverse(x*Q), n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) -1 = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-x)^n.
(2) -1 = Sum_{n>=0} A(x)^(n*(n+1)/2) * (1 - x^(2*n+1)) / (-x)^n.
(3) 1/x = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-1/x)^n.
(4) 1/x = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n*x) * (1 - A(x)^(n-1)/x).
(5) A(x) = Series_Reversion( x*Q(x) ), where Q(x) is the g.f. of A268301 and satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n*Q(x)) * (1 - x^(n-1)/Q(x)).
(6) x = Sum_{n>=1} a(n) * x^n * Q(x)^n, where Q(x) = Sum_{n>=0} A268301(n)/2*(x/4)^n.
a(n) ~ c * d^n / n^(3/2), where d = 24.827421130209954998234265953843191542003179657... and c = 0.020386712793003585674903530668163000681070027... . - Vaclav Kotesovec, Mar 02 2016

A355350 G.f. A(x,y) satisfies: xy = 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, 1, 0, 9, 6, 1, 0, 22, 27, 10, 1, 0, 51, 98, 66, 15, 1, 0, 108, 315, 340, 135, 21, 1, 0, 221, 918, 1495, 910, 246, 28, 1, 0, 429, 2492, 5838, 5070, 2086, 413, 36, 1, 0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1, 0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1, 0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1

Offset: 0

Paul D. Hanna, Jun 29 2022

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.
A355351(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A355352(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A355353(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A355354(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.
A355355(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.
A355356(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).
A355357(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0.
A354658(n) = T(2*n,n) for n >= 0 (central terms of this triangle).
Conjectures:
(C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds;
(C.2) Column 2 equals A023005, the number of partitions into parts of 6 kinds.

			G.f.: A(x,y) = 1 + x*y + x^2*(3*y + y^2) + x^3*(9*y + 6*y^2 + y^3) + x^4*(22*y + 27*y^2 + 10*y^3 + y^4) + x^5*(51*y + 98*y^2 + 66*y^3 + 15*y^4 + y^5) + x^6*(108*y + 315*y^2 + 340*y^3 + 135*y^4 + 21*y^5 + y^6) + x^7*(221*y + 918*y^2 + 1495*y^3 + 910*y^4 + 246*y^5 + 28*y^6 + y^7) + x^8*(429*y + 2492*y^2 + 5838*y^3 + 5070*y^4 + 2086*y^5 + 413*y^6 + 36*y^7 + y^8) + x^9*(810*y + 6372*y^2 + 20805*y^3 + 24543*y^4 + 14280*y^5 + 4284*y^6 + 652*y^7 + 45*y^8 + y^9) + x^10*(1479*y + 15525*y^2 + 68816*y^3 + 106535*y^4 + 83559*y^5 + 35168*y^6 + 8100*y^7 + 981*y^8 + 55*y^9 + y^10) + ...
where
x*y = ... - 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, given P(x) is the partition function (A000041),
x*y*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, 1];
n=3: [0, 9, 6, 1];
n=4: [0, 22, 27, 10, 1];
n=5: [0, 51, 98, 66, 15, 1];
n=6: [0, 108, 315, 340, 135, 21, 1];
n=7: [0, 221, 918, 1495, 910, 246, 28, 1];
n=8: [0, 429, 2492, 5838, 5070, 2086, 413, 36, 1];
n=9: [0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1];
n=10: [0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1];
n=11: [0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1];
n=12: [0, 4599, 81816, 630890, 1563705, 2033244, 1472261, 629280, 160965, 24145, 1991, 78, 1];
...
in which column 1 appears to equal A000716, the coefficients in P(x)^3,
and column 2 appears to equal 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 + ...

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

Cf. A355351 (row sums), A355352, A355353, A355354, A355355.
Cf. A355356, A355357, A354658 (central terms).
Cf. A354645, A354650 (related table), A000041, A000716, A023005.

{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( 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 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) x*y*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.

A354662 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2).

Original entry on oeis.org

1, 2, 6, 32, 190, 1236, 8482, 60434, 442788, 3315046, 25249888, 195040914, 1524256336, 12030033178, 95748941322, 767655502862, 6193902044684, 50257335231264, 409825115116030, 3356850545246400, 27606085924603602, 227850606781308660, 1886792409865105988

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 32*x^3 + 190*x^4 + 1236*x^5 + 8482*x^6 + 60434*x^7 + 442788*x^8 + 3315046*x^9 + 25249888*x^10 + ...
such that A = A(x) satisfies:
(1) 2 = ... + x^36*A^28 + x^28*A^21 - x^21*A^15 - x^15*A^10 + x^10*A^6 + x^6*A^3 - x^3*A - x + 1 + A - x*A^3 - x^3*A^6 + x^6*A^10 + x^10*A^15 - x^15*A^21 - x^21*A^28 + x^28*A^36 +--+ ...
(2) 2 = (1-x) + (1-x^3)*A - x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 + x^10*(1-x^11)*A^15 - x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(3) 2 = (1+A) - (1+A^3)*x - A*(1+A^5)*x^3 + A^3*(1+A^7)*x^6 + A^6*(1+A^9)*x^10 - A^10*(1+A^11)*x^15 - A^15*(1+A^13)*x^21 + A^21*(1+A^15)*x^28 + ...
(4) 2 = (1 + x*A)*(1 + A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 + x^2*A) * (1 + x^3*A^3)*(1 + x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 + x^4*A^3) * (1 + x^5*A^5)*(1 + x^4*A^5)*(1 - x^5*A^4) * ...

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

Cf. A354649, A354650, A354661, A354663, A354664, A268299, A354652, A354653, A354654.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 2 = Sum_{n=-oo..+oo}  (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) 2 = Sum_{n>=0}  (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) 2 = Sum_{n>=0}  (-x)^(n*(n+1)/2) * (1 + A(x)^(2*n+1)) * A(x)^(n*(n-1)/2).
(4) 2 = Product_{n>=1}  (1 - (-x)^n*A(x)^n) * (1 + (-x)^(n-1)*A(x)^n) * (1 + (-x)^n*A(x)^(n-1)), by the Jacobi triple product identity.
a(n) = (-1)^n * Sum_{k=0..2*n+1} A354649(n,k)*2^k, for n >= 0.
a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-2)^k, for n >= 0.

A354652 G.f. A(x) satisfies: -2 = Sum_{n=-oo..oo} (-1)^n * x^(n(n+1)/2) A(x)^(n*(n-1)/2).

Original entry on oeis.org

3, 26, 702, 24312, 964654, 41438412, 1876038114, 88154317378, 4258925591364, 210228411365958, 10556622328639744, 537564689914558410, 27693960347082015456, 1440798064785384773930, 75590961232091579641890, 3994794446280096850372038, 212460780898577846286309772

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = 3 + 26*x + 702*x^2 + 24312*x^3 + 964654*x^4 + 41438412*x^5 + 1876038114*x^6 + 88154317378*x^7 + 4258925591364*x^8 + ...
such that A = A(x) satisfies:
(1) -2 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
(2) -2 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(3) -2 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
(4) -2 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...

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

Cf. A354649, A354650, A268299, A354653, A354654, A354661, A354662, A354663, A354664.

(* Calculation of constant d: *) 1/r /. FindRoot[{r*s * QPochhammer[1/r, r*s] * QPochhammer[1/s, r*s] * QPochhammer[r*s] / ((-1 + r)*(-1 + s)) == -2, -2*(-1 + r)*(-1 + s)*Log[r*s] * Derivative[0, 1][QPochhammer][1/r, r*s] / QPochhammer[1/r, r*s] + r*s*Log[r*s] * QPochhammer[1/r, r*s] * QPochhammer[r*s, r*s] * Derivative[0, 1][QPochhammer][1/s, r*s] + (2*(-1 + r)*(QPochhammer[r*s, r*s]*(Log[r*s] + (-1 + s)*QPolyGamma[0, 1, r*s] - (-1 + s)* QPolyGamma[0, -Log[s]/Log[r*s], r*s]) - r*(-1 + s)*s*Log[r*s] * Derivative[0, 1][QPochhammer][ r*s, r*s])) / (r*s*QPochhammer[r*s, r*s]) == 0}, {r, 1/58}, {s, 4}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -2 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -2 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -2 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi triple product identity.
a(n) = (-1)^(n+1) * Sum_{k=0..2*n+1} A354649(n,k)*(-2)^k, for n >= 0.
a(n) = Sum_{k=0..2*n+1} A354650(n,k)*2^k, for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 58.4550529987436715308576941861333478321842744514285703940141525... and c = 0.757806366482059336893833755732847891553108700077361984219509... - Vaclav Kotesovec, Jun 08 2022
A(1/d) = 4.539776191075... where 1/d = 0.0171071609501661... and d is the value given above by Vaclav Kotesovec. - Paul D. Hanna, Jul 30 2022
Formula (4) can be rewritten as the functional equation QPochhammer(x*y) * QPochhammer(1/x, x*y)/(1 - 1/x) * QPochhammer(1/y, x*y)/(1 - 1/y) = -2. - Vaclav Kotesovec, Jan 19 2024

cp's OEIS Frontend

A354655 Column 2 of triangle A354650: a(n) = A354650(n,2), for n >= 1.

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A354658 A diagonal of triangle A354650: a(n) = A354650(n,n), for n >= 0.

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A354656 Column 3 of triangle A354650: a(n) = A354650(n,3), for n >= 1.

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A354659 A diagonal of triangle A354650: a(n) = A354650(n,n+1), for n >= 0.

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A354660 a(n) = A354650(n,2*n), for n >= 0.

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A354649 G.f. A(x,y) satisfies: y = f(x,A(x,y)), where f(x,y) = Sum_{n=-oo..oo} x^(n*(n+1)/2) * y^(n*(n-1)/2) is Ramanujan's theta function.

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A268299 G.f. A(x) satisfies: -1 = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n/x) * (1 - A(x)^(n-1)*x).

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Mathematica

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A355350 G.f. A(x,y) satisfies: 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.

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A354649 G.f. A(x,y) satisfies: y = f(x,A(x,y)), where f(x,y) = Sum_{n=-oo..oo} x^(n(n+1)/2) y^(n*(n-1)/2) is Ramanujan's theta function.

A355350 G.f. A(x,y) satisfies: xy = 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.

A354662 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2).

A354652 G.f. A(x) satisfies: -2 = Sum_{n=-oo..oo} (-1)^n * x^(n(n+1)/2) A(x)^(n*(n-1)/2).