A354652 -id:A354652 - cp's OEIS frontend

A354650 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.

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

Unsigned version of A354649.
Column 1 equals 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 A001764, with g.f. C(x) = 1 + x*C(x)^3.
T(n,1) = A000716(n), for n >= 0.
T(n,2) = A354655(n), for n >= 1.
T(n,3) = 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) = A354660(n), for n >= 0.
T(n,2*n+1) = A001764(n), for n >= 0.
Antidiagonal sums = A268650.
Row sums = A268299 (with offset).
Sum_{k=0..2*n+1} T(n,k)*2^k = A354652(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*3^k = A354653(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*4^k = A354654(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-1)^k = -A354661(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-2)^k = -A354662(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-3)^k = -A354663(n), for n >= 0.
Sum_{k=0..2*n+1} T(n,k)*(-4)^k = -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) + ...
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 A001764, with g.f. C(x) = 1 + x*C(x)^3.
Column 1 equals 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, A354649.

{T(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=0,12,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} (-1)^n * 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} (-1)^n * 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} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x,y)^(n*(n+1)/2).
(5) -y = Sum_{n>=0} (-1)^n * A(x,y)^(n*(n-1)/2) * (1 - A(x,y)^(2*n+1)) * x^(n*(n+1)/2).
Formulas for terms in rows.
(6) T(n,1) = A000716(n), the number of partitions of n into parts of 3 kinds.
(7) T(n,2*n+1) = A001764(n) = binomial(3*n,n)/(2*n+1), for n >= 0.

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.

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.

A354653 G.f. A(x) satisfies: -3 = Sum_{n=-oo..oo} (-1)^n * x^(n(n+1)/2) A(x)^(n*(n-1)/2).

Original entry on oeis.org

4, 63, 3024, 188688, 13492350, 1044853344, 85281392688, 7224776707896, 629288553814092, 56002675660109424, 5070000855941708292, 465454828626459320736, 43230859988456631732954, 4054827527508982869148392, 383529048423080768494135488, 36541031890621600233033859488

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = 4 + 63*x + 3024*x^2 + 188688*x^3 + 13492350*x^4 + 1044853344*x^5 + 85281392688*x^6 + 7224776707896*x^7 + 629288553814092*x^8 + ...
such that A = A(x) satisfies:
(1) -3 = ... + 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) -3 = (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) -3 = (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) -3 = (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, A354652, A354654, A354661, A354662, A354663, A354664.

{a(n) = my(A=[4]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(3 + 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) );A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -3 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -3 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -3 = 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)*(-3)^k, for n >= 0.
a(n) = Sum_{k=0..2*n+1} A354650(n,k)*3^k, for n >= 0.

A354654 G.f. A(x) satisfies: -4 = Sum_{n=-oo..oo} (-1)^n * x^(n(n+1)/2) A(x)^(n*(n-1)/2).

Original entry on oeis.org

5, 124, 9300, 912520, 102616748, 12498655200, 1604505393140, 213790010204692, 29287693334340840, 4099332312599011100, 583685111605968443456, 84277588096627459702860, 12310921909740521584887824, 1816058097888803062860159620, 270156262107594683175523302780

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = 5 + 124*x + 9300*x^2 + 912520*x^3 + 102616748*x^4 + 12498655200*x^5 + 1604505393140*x^6 + 213790010204692*x^7 + 29287693334340840*x^8 + ...
such that A = A(x) satisfies:
(1) -4 = ... + 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) -4 = (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) -4 = (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) -4 = (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, A354652, A354653, A354661, A354662, A354663, A354664.

{a(n) = my(A=[5]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(4 + 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) );A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -4 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -4 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -4 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -4 = 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)*(-4)^k, for n >= 0.
a(n) = Sum_{k=0..2*n+1} A354650(n,k)*4^k, for n >= 0.

A354661 G.f. A(x) satisfies: 1 = Sum_{n=-oo..oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2), with A(0) = 0.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 8, 0, 0, 44, 0, 6, 280, 0, 96, 1934, 0, 1124, 14088, 18, 11792, 106536, 648, 117626, 828360, 13416, 1142288, 6580780, 216000, 10921088, 53184864, 3019614, 103408416, 435930008, 38629656, 973041448, 3615741192, 465419760, 9118011128, 30298375236

Offset: 1

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = x + 2*x^4 + 8*x^7 + 44*x^10 + 6*x^12 + 280*x^13 + 96*x^15 + 1934*x^16 + 1124*x^18 + 14088*x^19 + 18*x^20 + 11792*x^21 + ...
such that A = A(x) satisfies:
(1) 1 = ... + 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) 1 = (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) 1 = (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) 1 = (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 = 1..400
Vaclav Kotesovec, Plot of a(n+1)/a(n) for n = 18..400

Cf. A354649, A354650, A354662, A354663, A354664, A268299, A354652, A354653, A354654.

{a(n) = my(A=[0]); for(i=0,n, A = concat(A,0);
A[#A] = -polcoeff(-1 + 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;A[n+1]}
for(n=1,50,print1(a(n),", "))

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

A354663 G.f. A(x) satisfies: 3 = Sum_{n=-oo..oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2).

Original entry on oeis.org

2, 9, 108, 1848, 36306, 771768, 17280096, 401451192, 9587095686, 233892105912, 5804193409056, 146051807458320, 3717875447707254, 95571022734750600, 2477365983601721280, 64684289495622383472, 1699638032224106092368, 44909438746576707103608

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = 2 + 9*x + 108*x^2 + 1848*x^3 + 36306*x^4 + 771768*x^5 + 17280096*x^6 + 401451192*x^7 + 9587095686*x^8 + 233892105912*x^9 + ...
such that A = A(x) satisfies:
(1) 3 = ... + 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) 3 = (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) 3 = (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) 3 = (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, A354662, A354664, A268299, A354652, A354653, A354654.

{a(n) = my(A=[2]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff(-3 + 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;A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 3 = Sum_{n=-oo..oo}  (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) 3 = Sum_{n>=0}  (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) 3 = Sum_{n>=0}  (-1)^(n*(n+1)/2) * A(x)^(n*(n-1)/2) * (1 + A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) 3 = 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)*3^k, for n >= 0.
a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-3)^k, for n >= 0.

A354664 G.f. A(x) satisfies: 4 = Sum_{n=-oo..oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2).

Original entry on oeis.org

3, 28, 756, 28200, 1205228, 55731456, 2714642292, 137199520340, 7127794098792, 378292284479388, 20421818573265728, 1117886561607128940, 61904487399635790288, 3461693986652051482948, 195203095905903229325340, 11087371481682320212435332, 633751222047605882649272600

Offset: 0

Paul D. Hanna, Jun 02 2022

nonn

			G.f.: A(x) = 3 + 28*x + 756*x^2 + 28200*x^3 + 1205228*x^4 + 55731456*x^5 + 2714642292*x^6 + 137199520340*x^7 + 7127794098792*x^8 + ...
such that A = A(x) satisfies:
(1) 4 = ... + 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) 4 = (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) 4 = (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) 4 = (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, A354662, A354663, A268299, A354652, A354653, A354654.

(* 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)) == 4, -4*(Log[-r*s] - (1 + s)*QPolyGamma[0, 1, -r*s] + (1 + s) * QPolyGamma[0, -Log[-s]/Log[-r*s], -r*s]) / (s*Log[-r*s]) + 4*r*(1 + s) * Derivative[0, 1][QPochhammer][1/r, -r*s] / QPochhammer[1/r, -r*s] + r^2*s*QPochhammer[1/r, -r*s]*QPochhammer[-r*s] * Derivative[0, 1][QPochhammer][-1/s, -r*s]/(-1 + r) + 4*r*(1 + s)*Derivative[0, 1][QPochhammer][-r*s, -r*s] / QPochhammer[-r*s] == 0}, {r, 1/50}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)

{a(n) = my(A=[3]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff(-4 + 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;A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 4 = Sum_{n=-oo..oo}  (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) 4 = Sum_{n>=0}  (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) 4 = Sum_{n>=0}  (-1)^(n*(n+1)/2) * A(x)^(n*(n-1)/2) * (1 + A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) 4 = 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)*4^k, for n >= 0.
a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-4)^k, for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 62.81220628370975097276726417958831026998790927499386157136003... and c = 0.71771306470564419436314253512374835316192083855385416486... - Vaclav Kotesovec, Jun 08 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) = 4. - Vaclav Kotesovec, Jan 19 2024

cp's OEIS Frontend

A354650 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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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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A354662 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

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A354653 G.f. A(x) satisfies: -3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

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A354654 G.f. A(x) satisfies: -4 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

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A354661 G.f. A(x) satisfies: 1 = Sum_{n=-oo..oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2), with A(0) = 0.

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A354663 G.f. A(x) satisfies: 3 = Sum_{n=-oo..oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

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A354664 G.f. A(x) satisfies: 4 = Sum_{n=-oo..oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

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A354650 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.

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.

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

A354653 G.f. A(x) satisfies: -3 = Sum_{n=-oo..oo} (-1)^n * x^(n(n+1)/2) A(x)^(n*(n-1)/2).

A354654 G.f. A(x) satisfies: -4 = Sum_{n=-oo..oo} (-1)^n * x^(n(n+1)/2) A(x)^(n*(n-1)/2).

A354661 G.f. A(x) satisfies: 1 = Sum_{n=-oo..oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2), with A(0) = 0.

A354663 G.f. A(x) satisfies: 3 = Sum_{n=-oo..oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2).

A354664 G.f. A(x) satisfies: 4 = Sum_{n=-oo..oo} (-x)^(n(n+1)/2) A(x)^(n*(n-1)/2).