A205968
a(n) = Fibonacci(n)*A008655(n) for n >= 1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x) + theta_2(x)*theta_2(3*x))^4.
Original entry on oeis.org
1, 24, 216, 1776, 5256, 15120, 63936, 107328, 294840, 823344, 1496880, 2845152, 9334656, 12291216, 28012608, 68251680, 110883528, 188343792, 563167296, 688359840, 1493387280, 3343696512, 5095667232, 8368761024, 24087248640, 28361250600, 57633511824, 128471514096
Offset: 0
G.f.: A(x) = 1 + 24*x + 216*x^2 + 1776*x^3 + 5256*x^4 + 15120*x^5 + ...
where A(x) = 1 + 1*24*x + 1*216*x^2 + 2*888*x^3 + 3*1752*x^4 + 5*3024*x^5 + ... + Fibonacci(n)*A008655(n)*x^n + ...
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A008655 := CoefficientList[Series[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, 250}], q]; b := Table[A008655[[n]], {n, 1, 120}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jul 16 2018 *)
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + sum(m=1,n, 24*fibonacci(m)*m^3*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*fibonacci(3*m)*(3*m)^3*x^(3*m)/(1-Lucas(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ),n)}
for(n=0,40,print1(a(n),", "))
A209448
a(n) = Pell(n)*A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x)+theta_2(x)*theta_2(3*x))^4.
Original entry on oeis.org
1, 24, 432, 4440, 21024, 87696, 559440, 1395264, 5728320, 23852760, 64719648, 183528288, 898460640, 1765134672, 6002425728, 21820957200, 52895150208, 134056553904, 598084104240, 1090757945760, 3530801969856, 11795485116480, 26821191064896, 65724336729792
Offset: 0
G.f.: A(x) = 1 + 24*x + 432*x^2 + 4440*x^3 + 21024*x^4 + 87696*x^5 +...
where A(x) = 1 + 1*24*x + 2*216*x^2 + 5*888*x^3 + 12*1752*x^4 + 29*3024*x^5 +...+ Pell(n)*A008655(n)*x^n +...
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A008655[n_]:= SeriesCoefficient[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, n}]; b:= Table[A008655[n], {n, 0, 102}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n, 2]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 26 2018 *)
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{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 + sum(m=1,n, 24*Pell(m)*m^3*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*Pell(3*m)*(3*m)^3*x^(3*m)/(1-A002203(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ),n)}
for(n=0,40,print1(a(n),", "))
A341306
Fourier coefficients of the modular form F_{3A}^8.
Original entry on oeis.org
1, 48, 1008, 12144, 92784, 473760, 1706544, 4818048, 12317040, 29078832, 59093280, 114031296, 219429552, 367093536, 621859968, 1037221920, 1583864688, 2403178848, 3747390192, 5232056640, 7550261280, 10938344064, 14714951616, 19930041216, 28075097520, 35731471440
Offset: 0
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A341306 := proc(n)
add( A004016(i)*x^i,i=0..n) ;
coeftayl(%^8,x=0,n) ;
end proc:
seq(A341306(n),n=0..25) ; # R. J. Mathar, Feb 22 2021
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A004016[n_] := If[n == 0, 1, 6 Sum[KroneckerSymbol[d, 3], {d, Divisors[n]}]];
a[n_] := SeriesCoefficient[Sum[A004016[i]*x^i, {i, 0, n}]^8, {x, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023, after R. J. Mathar *)
A341556
Fourier coefficients of the modular form F_{3A}^12.
Original entry on oeis.org
1, 72, 2376, 47592, 646344, 6305904, 45821160, 255215808, 1125009864, 4097478600, 12975540336, 37101202848, 96867424872, 232791251760, 526183909056, 1128351033648, 2286328733640, 4451427831312, 8386379869896, 15130456297056, 26613241744752, 45684436953024, 75935115663264
Offset: 0
A341557
Fourier coefficients of the modular form (1/t_{3A}) * F_{3A}^12.
Original entry on oeis.org
0, 1, 30, 333, 1444, -570, -21114, -22576, 121848, 64233, -276300, 589260, -1198764, 133766, -957216, 2920590, 2491792, -1616958, 1647054, -5312428, -14819880, -4158576, 20300904, 16879848, 19051416, -22583225, 38165172, -81066987, -47716288, 66118494, 370980, -51834232
Offset: 0
A341561
Fourier coefficients of the modular form (1/t_{3A}) * F_{3A}^16.
Original entry on oeis.org
0, 1, 54, 1269, 16804, 134406, 628398, 1311968, -1701864, -14345991, -16443324, 25426764, 11246580, 16601078, 505866816, -113853762, -1326884336, 1507092642, -3873575034, 100819028, 2685180888, 6885133920, -20849400, 10111254408, -10371867912, -412371305, -58625773596
Offset: 0
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def a(n):
if n==0: return 0
eta = x^(1/24)*product([(1 - x^k) for k in range(1, n)])
t3A = ((eta/eta(x=x^3))^12 + 27)^2/(eta/eta(x=x^3))^12
F3A = sum([rising_factorial(1/6, k)*rising_factorial(1/3, k)/
(rising_factorial(1,k)^2)*(108/t3A)^k for k in range(n)])
f = F3A^16/t3A
return f.taylor(x,0,n).coefficients()[n-1][0] # Robin Visser, Jul 23 2023
Showing 1-6 of 6 results.
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