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A354649
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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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17
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-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
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OFFSET
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0,4
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COMMENTS
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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,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)).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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