cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Comments

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)).

Examples

			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.

Links

Crossrefs

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.

Programs

  • PARI
    {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(""))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Comments

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)).

Examples

			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.

Links

Crossrefs

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.

Programs

  • PARI
    {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(""))

Formula

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

Views

Author

Paul D. Hanna, Feb 26 2016

Keywords

Comments

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

Examples

			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)) *...

Links

Crossrefs

Cf. A354650, A268300, A268301, A268650.

Programs

  • Mathematica
    (* 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 *)
  • PARI
    {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), ", "))

Formula

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

A268651 G.f. A(x) satisfies: 1 = Product_{n>=1} (1 - x^n) * (1 - x^(n+1)/A(x)) * (1 - x^(n-2)*A(x)).

Original entry on oeis.org

1, 1, 2, 5, 9, 22, 52, 146, 377, 1036, 2810, 8014, 22790, 66100, 191541, 562926, 1660975, 4944766, 14767136, 44357952, 133698623, 404810569, 1229434572, 3746595869, 11447723074, 35075829156, 107724187826, 331605018200, 1022842337041, 3161156987190, 9787096605716, 30352665554591, 94279407445012, 293277650593792, 913565090912339, 2849489942324466, 8898714901181309, 27822251614174021, 87083081436755770
Offset: 0

Views

Author

Paul D. Hanna, Mar 02 2016

Keywords

Comments

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.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 22*x^5 + 52*x^6 + 146*x^7 + 377*x^8 + 1036*x^9 + 2810*x^10 + 8014*x^11 + 22790*x^12 + 66100*x^13 + 191541*x^14 + 562926*x^15 +...
where A(x) satisfies the Jacobi Triple Product:
1 = (1-x)*(1-x^2/A(x))*(1-1/x*A(x)) * (1-x^2)*(1-x^3/A(x))*(1-1*A(x)) * (1-x^3)*(1-x^4/A(x))*(1-x*A(x)) * (1-x^4)*(1-x^5/A(x))*(1-x^2*A(x)) * (1-x^5)*(1-x^6/A(x))*(1-x^3*A(x)) * (1-x^6)*(1-x^7/A(x))*(1-x^4*A(x)) *...
Also
x = (A(x)-1)*A(x) - x*(A(x)^3-1) + x^3*(A(x)^5-1)/A(x) - x^6*(A(x)^7-1)/A(x)^2 + x^10*(A(x)^9-1)/A(x)^3 - x^15*(A(x)^11-1)/A(x)^4 + x^21*(A(x)^13-1)/A(x)^5 +...

Links

Crossrefs

Cf. A268650.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{s*QPochhammer[r, r] * QPochhammer[r/s, r] * QPochhammer[s/r^2, r] == (s - r)*(1 - s/r^2), (r^3 - s^2)* Log[r] + (r^3 - r*s - r^2*s + s^2) * (QPolyGamma[0, Log[r/s]/Log[r], r] - QPolyGamma[0, Log[s/r^2]/Log[r], r]) == 0}, {r, 1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
  • PARI
    {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); A[#A]=-Vec( sum(m=1,sqrtint(2*#A)+2,(-1)^m*(x*Ser(A))^(m*(m-1)/2)*(1-x^(2*m-1))/x^m) )[#A-1] );Vec(x/serreverse(x*Ser(A)))[n+1]}
for(n=0,40,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..oo} (-1)^n * x^((n-1)*(n-2)/2) * A(x)^n.
(2) x = sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (A(x)^(2*n+1) - 1) / A(x)^(n-1).
(3) A(x) = x / Series_Reversion( G(x) ), where G(x) is the g.f. of A268650.
a(n) ~ c * d^n / n^(3/2), where d = 3.25766000970998791... and c = 0.661369655158037... . - Vaclav Kotesovec, Mar 05 2016

A354647 G.f. A(x) satisfies: -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 1, 3, 9, 25, 78, 256, 881, 3064, 10831, 38766, 140550, 514625, 1900301, 7067013, 26448613, 99539716, 376489459, 1430330451, 5455742957, 20885223619, 80213926069, 309002022843, 1193616950854, 4622372591972, 17942238661229, 69795082381496, 272046051362013
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2022

Keywords

Examples

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 9*x^4 + 25*x^5 + 78*x^6 + 256*x^7 + 881*x^8 + 3064*x^9 + 10831*x^10 + 38766*x^11 + 140550*x^12 + ...
such that A = A(x) satisfies:
(1) -x^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) -x^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) -x^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) -x^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) * ...

Crossrefs

Cf. A268650, A354648, A354649.

Programs

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

Formula

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

A354648 G.f. A(x) satisfies: -x^3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 3, 9, 22, 54, 135, 368, 1060, 3135, 9295, 27472, 81309, 242255, 728429, 2208483, 6736523, 20634196, 63410076, 195467757, 604457802, 1875053982, 5833449236, 18195767301, 56888745654, 178238369769, 559538565187, 1759796017533, 5544359742297
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2022

Keywords

Examples

			G.f.: A(x) = 1 + x^3 + 3*x^4 + 9*x^5 + 22*x^6 + 54*x^7 + 135*x^8 + 368*x^9 + 1060*x^10 + 3135*x^11 + 9295*x^12 + 27472*x^13 + ...
such that A = A(x) satisfies:
(1) -x^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) -x^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) -x^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) -x^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) * ...

Crossrefs

Cf. A268650, A354647, A354649, A000716.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(x^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),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -x^3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -x^3 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -x^3 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -x^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.
Showing 1-6 of 6 results.

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