OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2200 from Paul D. Hanna)
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) A(x) = (1/3) * Sum_{n>=0} Product_{k=0..2*n} (x^k + x^n + x^(2*n-k)).
(2) A(x) = Sum_{n>=0} x^n * Product_{k=0..n-1} (x^k + x^n + x^(2*n-k))^2.
(3) A(x) = (1/3) * Sum_{n>=0} x^(n*(2*n+1)) * Product_{k=0..2*n} (1 + x^(n-k) + x^(2*n-2*k)).
(4) A(x) = (1/3) * Sum_{n>=0} x^(n*(2*n+1)) * Product_{k=0..2*n} (1/x^(n-k) + 1 + x^(n-k)).
(5) A(x) = (1/3) * Sum_{n>=0} x^(n*(2*n+1)) * Product_{k=0..2*n} (1 + 1/x^(n-k) + 1/x^(2*n-2*k)).
(6) A(x) = (1/3) * Sum_{n>=0} x^(2*n*(2*n+1)) * Product_{k=0..2*n} (1/x^k + 1/x^n + 1/x^(2*n-k)).
From Paul D. Hanna, Oct 09 2024: (Start)
(7) A(x) = Sum_{n>=0} x^(n^2) * Product_{k=1..n} (1 - x^(3*k))^2 / (1 - x^k)^2.
(8) A(x) = Sum_{n>=0} x^(n^2) * Product_{k=1..n} (1 + x^k + x^(2*k))^2.
(End)
a(n) ~ c * d^sqrt(n) / sqrt(n), where d = A376152 = 4.9880208766009... and c = sqrt(1/54 + 5*cosh(arccosh(7*sqrt(11/2)/16)/3)/(27*sqrt(22))) = 0.241068202175... - Vaclav Kotesovec, Sep 28 2024, updated Oct 09 2024
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 10*x^8 + 11*x^9 + 15*x^10 + 15*x^11 + 18*x^12 + 20*x^13 + 25*x^14 + 30*x^15 + 38*x^16 + 47*x^17 + 57*x^18 + 67*x^19 + 78*x^20 + ...
where
A(x) = 1 + (1 + x + x^2)*(x)*(x^2 + x + 1) + (1 + x^2 + x^4)*(x + x^2 + x^3)*(x^2)*(x^3 + x^2 + x)*(x^4 + x^2 + 1) + (1 + x^3 + x^6)*(x + x^3 + x^5)*(x^2 + x^3 + x^4)*(x^3)*(x^4 + x^3 + x^2)*(x^5 + x^3 + x)*(x^6 + x^3 + 1) + (1 + x^4 + x^8)*(x + x^4 + x^7)*(x^2 + x^4 + x^6)*(x^3 + x^4 + x^5)*(x^4)*(x^5 + x^4 + x^3)*(x^6 + x^4 + x^2)*(x^7 + x^4 + x)*(x^8 + x^4 + 1) + (1 + x^5 + x^10)*(x + x^5 + x^9)*(x^2 + x^5 + x^8)*(x^3 + x^5 + x^7)*(x^4 + x^5 + x^6)*(x^5)*(x^6 + x^5 + x^4)*(x^7 + x^5 + x^3)*(x^8 + x^5 + x^2)*(x^9 + x^5 + x)*(x^10 + x^5 + 1) + ...
Also,
A(x) = 1 + x*(1 + x + x^2)^2 + x^2*(1 + x^2 + x^4)^2*(x + x^2 + x^3)^2 + x^3*(1 + x^3 + x^6)^2*(x + x^3 + x^5)^2*(x^2 + x^3 + x^4)^2 + x^4*(1 + x^4 + x^8)^2*(x + x^4 + x^7)^2*(x^2 + x^4 + x^6)^2*(x^3 + x^4 + x^5)^2 + x^5*(1 + x^5 + x^10)^2*(x + x^5 + x^9)^2*(x^2 + x^5 + x^8)^2*(x^3 + x^5 + x^7)^2*(x^4 + x^5 + x^6)^2 + ...
SPECIFIC VALUES.
A(t) = 7 at t = 0.66668704736936585046859672241821389017558257705439339...
A(t) = 6 at t = 0.64513809385910573788372368634772347751697803188552164...
A(t) = 5 at t = 0.61639010238633204213526430692013003520814209008383800...
A(t) = 4 at t = 0.57545188136244196253678514659912022278976129786049251...
A(t) = 3 at t = 0.51093469574142600352566002004049869356160992832828805...
A(t) = 2 at t = 0.38925040919555545279428903616909363335667114006118874...
A(4/5) = 33.86295094486999840248628061724081807284197309832190750...
A(3/4) = 15.71390570183068296805142809300098703963996686273128437...
A(2/3) = 6.998922814611911009050207691553160959950411531472265898...
A(3/5) = 4.551745873136373778485262039216993578932737039944687958...
A(1/2) = 2.87450225671651109577680741009657439874438592581613285485257...
where A(1/2) = 1 + 7^2/2^5 + 147^2/2^16 + 10731^2/2^33 + 2929563^2/2^56 + 3096548091^2/2^85 + 12884736606651^2/2^120 + 212765655585627963^2/2^161 + 13998490777945220569659^2/2^208 + ... + A376227(n)^2/2^(n*(3*n+2)) + ..., where A376227(n) = Product_{k=1..n} (1 + 2^k + 2^(2*k)).
A(2/5) = 2.062036845797808963480254546496778756663866279595140073...
A(1/3) = 1.728128514830894263417956669231253604769749542061786338...
A(1/4) = 1.438324287250845860310741641820056491309903730120221376...
A(1/5) = 1.310189970721194144762503370434773514855060963388422496...
MATHEMATICA
nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1 + x^j + x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 28 2024 *)
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k + x^(2*k))*(1 + x^k + x^(2*k)) * x^(2*k-1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Oct 08 2024 *)
PROG
(PARI) {a(n) = my(A = (1/3)*sum(m=0, n, prod(k=0, 2*m, x^k + x^m + x^(2*m-k) +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Sep 27 2024
STATUS
approved