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 1-3 of 3 results.

A370034 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 4, 15, 53, 185, 711, 3270, 17297, 95108, 511258, 2653139, 13479835, 68633758, 356913516, 1906525759, 10388550830, 57084621325, 313692565172, 1719365476703, 9416232699651, 51699722653269, 285294478988749, 1583233662850172, 8826549215612727, 49354550054780111, 276444281747417079
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 4*x^2 + 15*x^3 + 53*x^4 + 185*x^5 + 711*x^6 + 3270*x^7 + 17297*x^8 + 95108*x^9 + 511258*x^10 + 2653139*x^11 + 13479835*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*x - 2*x^4 - 2*x^9 - 2*x^16 - 2*x^25 - 2*x^36 - 2*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05211271680112049721451382589099198923178830298930738503...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A)^n = 2 - Pi^(1/4)/gamma(3/4) = 0.913565188786691985...
(V.2) Let A = A(exp(-2*Pi)) = 0.001881490436109324727231096204943046774873234177072692211...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 4*A)^n = 2 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.99626511451226...
(V.3) Let A = A(-exp(-Pi)) = -0.03679381086518350821622244996144281973183248006035375080...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 4*A)^n = 2 - (Pi/2)^(1/4)/gamma(3/4) = 1.08642086184388317...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001853590408074327278987912837104527635895010708605840824...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 4*A)^n = 2 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.003734885439...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370035, A370036, A370037, A370038, A370039, A370043.
Cf. A369671.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 4*Ser(A))^m ) - 1 + 2*sum(m=1,#A, x^(m^2) ), #A-1)/4 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 4*A(x))^(n-1) = 1 - 2*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 4*x^n*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 4*x^n*A(x))^(n+1) = 1 - 2*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.

A369672 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x).

Original entry on oeis.org

1, -4, 19, -100, 569, -3416, 21302, -136636, 895572, -5971096, 40366463, -276036720, 1905940182, -13269019988, 93040431283, -656472509864, 4657492107245, -33205607204468, 237777067846451, -1709374453370956, 12332468208675821, -89262196983781332, 647988910138661556
Offset: 1

Views

Author

Paul D. Hanna, Feb 03 2024

Keywords

Comments

Note: theta_3(x) = Sum_{n=-oo..+oo} x^(n^2) - see A000122.
Congruences:
(C.1) a(2*n) == 0 (mod 4) for n >= 1.
(C.2) a(n) == A369671(n) (mod 4) for n >= 1.
(C.3) a(2*n)/4 == -A369671(2*n)/4 (mod 4) for n >= 1.

Examples

			G.f.: A(x) = x - 4*x^2 + 19*x^3 - 100*x^4 + 569*x^5 - 3416*x^6 + 21302*x^7 - 136636*x^8 + 895572*x^9 - 5971096*x^10 + 40366463*x^11 - 276036720*x^12 + ...
where Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x), and
theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + ... + x^(n^2) + ...
RELATED SERIES.
When we break up the doubly infinite sum into the following parts
P = Sum_{n>=0} (-1)^n * (x^n - 4*A(x))^n = 1 + 3*x + 4*x^3 - 15*x^4 + 92*x^5 - 528*x^6 + 3196*x^7 - 20032*x^8 + 128819*x^9 - 845312*x^10 + 5638568*x^11 - 38122176*x^12 + ...
N = Sum_{n>=1} (-1)^n * x^(n^2) / (1 - 4*x^n*A(x))^n = -x - 4*x^3 + 17*x^4 - 92*x^5 + 528*x^6 - 3196*x^7 + 20032*x^8 - 128817*x^9 + 845312*x^10 - 5638568*x^11 + 38122176*x^12 + ...
we see that the sum equals P + N = theta_3(x).
SPECIAL VALUES.
(V.1) A(exp(-Pi)) = 0.036996905719511834010608252452763733693844226179196126014832...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) - 4*A(exp(-Pi)))^n = Pi^(1/4)/gamma(3/4) = 1.0864348112133080...
(V.2) A(exp(-2*Pi)) = 0.0018536158947374219405603135305712038712234615914707006019...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) - 4*A(exp(-2*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt(2 + sqrt(2))/2 = 1.0037348854877390...
(V.3) A(exp(-3*Pi)) = 0.0000806734779029429093753810781078431328279003228392603227...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-3*n*Pi) - 4*A(exp(-3*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt(1 + sqrt(3))/(108)^(1/8) = 1.000161399035140...
(V.4) A(exp(-4*Pi)) = 0.0000034872937107879617892620501277220047637185282553554945...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-4*n*Pi) - 4*A(exp(-4*Pi)))^n = Pi^(1/4)/gamma(3/4) * (2 + 8^(1/4))/4 = 1.000161399035140...
(V.5) A(exp(-5*Pi)) = 0.0000001507016366950287572418174619564191722052174968450159...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-5*n*Pi) - 4*A(exp(-5*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt((2 + sqrt(5))/5) = 1.0000003014034550...

Links

Crossrefs

Cf. A369671 (dual), A000122 (theta_3), A355868.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
A[#A] = polcoeff( (sum(n=-M,M, x^(n^2)) - sum(n=-#A,#A, (-1)^n * (x^n - 4*x*Ser(A))^n) )/4, #A); ); A[n]}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
A[#A] = polcoeff( (sum(n=-M,M, x^(n^2)) - sum(n=-#A,#A, (-1)^n * x^(n^2)/(1 - 4*x^(n+1)*Ser(A))^n) )/4, #A); ); A[n]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = Sum_{n=-oo..+oo} x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^(n-1) = Sum_{n=-oo..+oo} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 - 4*x^n*A(x))^n = Sum_{n=-oo..+oo} x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 - 4*x^n*A(x))^(n+1) = Sum_{n=-oo..+oo} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.
a(n) ~ c * (-1)^(n+1) * d^n / n^(3/2), where d = 7.7471235933114571108403244715948697607... and c = 0.26329435412874059034137968338302672... - Vaclav Kotesovec, Feb 03 2024

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 1-3 of 3 results.

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