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.

Showing 1-2 of 2 results.

A369682 Expansion of g.f. A(x) satisfying Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = theta_2(x^(1/2)) / x^(1/8).

Original entry on oeis.org

1, 4, 12, 38, 112, 332, 972, 2818, 8098, 23096, 65418, 184194, 516080, 1440334, 4008442, 11135682, 30912896, 85835538, 238601354, 664447912, 1854592214, 5189848462, 14561237108, 40954656118, 115428662380, 325847049200, 920772219740, 2602948470362, 7356994944096, 20779322594048
Offset: 0

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Author

Paul D. Hanna, Feb 04 2024

Keywords

Comments

Note: theta_2(x^(1/2)) / x^(1/8) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) - see A089799.

Examples

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 38*x^3 + 112*x^4 + 332*x^5 + 972*x^6 + 2818*x^7 + 8098*x^8 + 23096*x^9 + 65418*x^10 + 184194*x^11 + 516080*x^12 + ...
By definition, A = A(x) satisfies the sum of products
theta_2(x^(1/2))/x^(1/8) = (1 + A) - x*(1 + A)*(x + A) + x^2*(1 + A)*(x + A)*(x^2 + A) - x^3*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A) + x^4*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A) -+ ...
also, A = A(x) satisfies another sum of products
x*theta_2(x^(1/2))/x^(1/8) = 1 - 1/(1 + x*A) + x/((1 + x*A)*(1 + x^2*A)) - x^3/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)) + x^6/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)) - x^10/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)*(1 + x^5*A)) +- ...
Further, A = A(x) satisfies the continued fraction given by
theta_2(x^(1/2))/x^(1/8) = (1 + A)/(1 + x*(x + A)/(1 - x*(x + A) + x*(x^2 + A)/(1 - x*(x^2 + A) + x*(x^3 + A)/(1 - x*(x^3 + A) + x*(x^4 + A)/(1 - x*(x^4 + A) + x*(x^5 + A)/(1 - x*(x^5 + A) + ...))))))
where theta_2(x^(1/2))/x^(1/8) = 2 + 2*x + 2*x^3 + 2*x^6 + 2*x^10 + 2*x^15 + 2*x^21 + ... + 2*x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A369683, A369684, A089799 (theta_2).

Programs

  • PARI
    {a(n) = my(A=[1], M = sqrtint(2*n)+1); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-M,M, x^(n*(n+1)/2) ) - sum(n=0,#A, (-1)^n * x^n * prod(k=0,n, x^k + Ser(A)) ), #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 the following formulas.
(1) Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = Sum_{n=-oo..+oo} x^(n*(n+1)/2).
(2) Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) / Product_{k=1..n} (1 + x^k*A(x)) = x * Sum_{n=-oo..+oo} x^(n*(n+1)/2).
(3) theta_2(x^(1/2))/x^(1/8) = (1 + A(x))/(1 + F(1)), where F(n) = x*(x^n + A(x))/(1 - x*(x^n + A(x)) + F(n+1)), a continued fraction.
(4) x * theta_2(x^(1/2))/x^(1/8) = 1/(1 + F(1)), where F(n) = x^(n-1)/(1 - x^(n-1) + x^n*A + (1 + x^n*A) * F(n+1)), a continued fraction.

A369684 Expansion of g.f. A(x) satisfying Sum_{n>=0} x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = theta_4(x).

Original entry on oeis.org

1, -4, 6, -14, 27, -54, 110, -217, 445, -905, 1863, -3858, 7986, -16599, 34438, -71445, 148075, -306551, 634469, -1312707, 2716636, -5624353, 11649994, -24144393, 50059148, -103820127, 215351391, -446713118, 926604822, -1921881919, 3985904949, -8266207127, 17142752984
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2024

Keywords

Comments

Note: theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2) - see A002448.

Examples

			G.f.: A(x) = 1 - 4*x + 6*x^2 - 14*x^3 + 27*x^4 - 54*x^5 + 110*x^6 - 217*x^7 + 445*x^8 - 905*x^9 + 1863*x^10 - 3858*x^11 + 7986*x^12 + ...
By definition, A = A(x) satisfies the sum of products
theta_4(x) = (x + A) + x*(x + A)*(x^3 + A) + x^2*(x + A)*(x^3 + A)*(x^5 + A) + x^3*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A) + x^4*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A)*(x^9 + A) + ...
also, A = A(x) satisfies another sum of products
1 + x*theta_4(x) = 1/(1 - x*A) + x^2/((1 - x*A)*(1 - x^3*A)) + x^6/((1 - x*A)*(1 - x^3*A)*(1 - x^5*A)) + x^12/((1 - x*A)*(1 - x^3*A)*(1 - x^5*A)*(1 - x^7*A)) + x^20/((1 - x*A)*(1 - x^3*A)*(1 - x^5*A)*(1 - x^7*A)*(1 - x^9*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
theta_4(x) = (x + A)/(1 - x*(x^3 + A)/(1 + x*(x^3 + A) - x*(x^5 + A)/(1 + x*(x^5 + A) - x*(x^7 + A)/(1 + x*(x^7 + A) - x*(x^9 + A)/(1 + x*(x^9 + A) - x*(x^11 + A)/(1 + ...))))))
where theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36 -+ ... + (-1)^n*2*x^(n^2) + ...
SPECIAL VALUES.
(V.1) A(exp(-Pi)) = 0.83730587071032100304130860721328123647753330074821532779...
where Sum_{n>=0} exp(-n*Pi) * Product_{k=0..n} (exp(-(2*k+1)*Pi) + A(exp(-Pi))) = (Pi/2)^(1/4)/gamma(3/4) = 0.91357913815611682140...
(V.2) A(exp(-2*Pi)) = 0.992551062280675678319013190897648447080249317782864483...
where Sum_{n>=0} exp(-2*n*Pi) * Product_{k=0..n} (exp(-2*(2*k+1)*Pi) + A(exp(-2*Pi))) = (Pi/2)^(1/4)/gamma(3/4) * 2^(1/8) = 0.99626511456090713578995...
(V.3) A(exp(-4*Pi)) = 0.99998605070354391051731649267915065106164831758249400...
where Sum_{n=-oo..+oo} exp(-4*n*Pi) * Product_{k=0..n} (exp(-4*(2*k+1)*Pi) + A(exp(-4*Pi))) = Pi^(1/4)/gamma(3/4) * (sqrt(2) + 1)^(1/4)/2^(7/16) = 0.99999302531528758200931...
(V.4) A(exp(-10*Pi)) = 0.9999999999999091559572670393411928665159764707765156...
where Sum_{n=-oo..+oo} exp(-10*n*Pi) * Product_{k=0..n} (exp(-10*(2*k+1)*Pi) + A(exp(-10*Pi))) = Pi^(1/4)/gamma(3/4) * 2^(7/8)/((5^(1/4) - 1)*sqrt(5*sqrt(5) + 5)) = 0.99999999999995457797863...

Links

Crossrefs

Cf. A369683, A369682, A369671, A002448 (theta_4).

Programs

  • PARI
    {a(n) = my(A=[1], M = sqrtint(2*n)+1); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-M,M, (-1)^n * x^(n^2) ) - sum(n=0,#A, x^n * prod(k=0,n, x^(2*k+1) + Ser(A)) ), #A-1) ); H=A; A[n+1]}
for(n=0,40, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) Sum_{n>=0} x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(2) Sum_{n>=0} x^(n*(n+1)) / Product_{k=0..n} (1 - x^(2*k+1)*A(x)) = 1 + x * Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(3) theta_4(x) = (x + A(x))/(1 + F(1)), where F(n) = -x*(x^(2*n+1) + A(x))/(1 + x*(x^(2*n+1) + A(x)) + F(n+1)), a continued fraction.
(4) 1 + x*theta_4(x) = 1/((1 - x*A(x))*(1 + F(1))), where F(n) = -x^(2*n) / (1 + x^(2*n) - x^(2*n+1)*A + (1 - x^(2*n+1)*A)*F(n+1)), a continued fraction.
Showing 1-2 of 2 results.

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