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

Author

Paul D. Hanna, Jun 02 2022

Keywords

• nonn
• tabf

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

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

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.