A354650 - OEIS

%I #10 Mar 28 2023 13:18:16

%S 1,1,0,3,3,1,0,9,27,30,15,3,0,22,147,340,390,246,83,12,0,51,630,2530,

%T 5070,5928,4284,1908,486,55,0,108,2295,14595,45450,83559,98910,78282,

%U 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

%N 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.

%C Unsigned version of A354649.

%C 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.

%C The rightmost border equals A001764, with g.f. C(x) = 1 + x*C(x)^3.

%C T(n,1) = A000716(n), for n >= 0.

%C T(n,2) = A354655(n), for n >= 1.

%C T(n,3) = A354656(n), for n >= 1.

%C T(n,n) = A354658(n), for n >= 0.

%C T(n,n+1) = A354659(n), for n >= 0.

%C T(n,2*n) = A354660(n), for n >= 0.

%C T(n,2*n+1) = A001764(n), for n >= 0.

%C Antidiagonal sums = A268650.

%C Row sums = A268299 (with offset).

%C Sum_{k=0..2*n+1} T(n,k)*2^k = A354652(n), for n >= 0.

%C Sum_{k=0..2*n+1} T(n,k)*3^k = A354653(n), for n >= 0.

%C Sum_{k=0..2*n+1} T(n,k)*4^k = A354654(n), for n >= 0.

%C Sum_{k=0..2*n+1} T(n,k)*(-1)^k = -A354661(n), for n >= 0.

%C Sum_{k=0..2*n+1} T(n,k)*(-2)^k = -A354662(n), for n >= 0.

%C Sum_{k=0..2*n+1} T(n,k)*(-3)^k = -A354663(n), for n >= 0.

%C Sum_{k=0..2*n+1} T(n,k)*(-4)^k = -A354664(n), for n >= 0.

%C SPECIFIC VALUES.

%C (1) A(x,y) = -exp(-Pi) at x = -exp(-Pi), y = -Pi^(1/4)/gamma(3/4).

%C (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.

%C (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.

%C (4) A(x,y) = -exp(-4*Pi) at x = -exp(-4*Pi), y = -Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.

%C (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)).

%H Paul D. Hanna, <a href="/A354650/b354650.txt">Table of n, a(n) for n = 0..10301</a>

%F G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..2*n+1} T(n,k)*y^k satisfies:

%F (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.

%F (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.

%F (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.

%F (4) -y = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x,y)^(n*(n+1)/2).

%F (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).

%F Formulas for terms in rows.

%F (6) T(n,1) = A000716(n), the number of partitions of n into parts of 3 kinds.

%F (7) T(n,2*n+1) = A001764(n) = binomial(3*n,n)/(2*n+1), for n >= 0.

%e 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) + ...

%e such that A = A(x,y) satisfies:

%e (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 + ...

%e (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) * ...

%e (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 + ...

%e (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 + ...

%e This triangle of coefficients of x^n*y^k in g.f. A(x,y) for n >= 0, k = 0..2*n+1, begins:

%e 1, 1;

%e 0, 3, 3, 1;

%e 0, 9, 27, 30, 15, 3;

%e 0, 22, 147, 340, 390, 246, 83, 12;

%e 0, 51, 630, 2530, 5070, 5928, 4284, 1908, 486, 55;

%e 0, 108, 2295, 14595, 45450, 83559, 98910, 78282, 41580, 14355, 2937, 273;

%e 0, 221, 7476, 70737, 319605, 849450, 1472261, 1757688, 1484451, 891890, 375442, 105930, 18109, 1428;

%e 0, 429, 22302, 301070, 1886010, 6878907, 16386636, 27205308, 32683680, 28981855, 19081854, 9258678, 3231514, 771225, 113220, 7752;

%e 0, 810, 62100, 1157820, 9729720, 46977378, 147584556, 324283068, 520974180, 628884300, 579226362, 409367712, 221218179, 90115620, 26879160, 5559408, 715122, 43263; ...

%e The rightmost border equals A001764, with g.f. C(x) = 1 + x*C(x)^3.

%e 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.

%o (PARI) {T(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);

%o 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) );

%o polcoeff(A[n+1],k,y)}

%o for(n=0,12,for(k=0,2*n+1,print1(T(n,k),", "));print(""))

%Y Cf. A000716 (column 1), A354655 (column 2), A354656 (column 3).

%Y Cf. A354658 (T(n,n)), A354659 (T(n,n+1)), A354660 (T(n,2*n)), A001764 (right border).

%Y Cf. A268299 (y=1), A354652 (y=2), A354653 (y=3), A354654 (y=4).

%Y Cf. A354661 (y=-1), A354662 (y=-2), A354663 (y=-3), A354664 (y=-4).

%Y Cf. A268650 (antidiagonal sums), A354657, A354649.

%K nonn,tabf

%O 0,4

%A _Paul D. Hanna_, Jun 02 2022