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.

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

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

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

Cf. A354649, A354650, A268299, A354652, A354654, A354661, A354662, A354663, A354664.

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

Cf. A354649, A354650, A268299, A354652, A354653, A354661, A354662, A354663, A354664.

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

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

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

Cf. A354649, A354650, A354661, A354662, A354664, A268299, A354652, A354653, A354654.

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Examples

			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) * ...

Links

Crossrefs

Cf. A354649, A354650, A354661, A354662, A354663, A268299, A354652, A354653, A354654.

Programs

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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354656.

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354659, A354660.

Programs

  • PARI
    {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),", "))

Formula

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
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