A322204 G.f.: exp( Sum_{n>=1} A322203(n)*x^n/n ), where A322203(n) is the coefficient of x^n*y^n/n in Sum_{n>=1} -log(1 - (x^n + y^n)).
1, 1, 3, 7, 20, 54, 168, 518, 1702, 5672, 19413, 67329, 236994, 842362, 3022320, 10924142, 39749219, 145457241, 534996370, 1976582432, 7332199623, 27298096431, 101968071485, 382033462335, 1435270419582, 5405847465772, 20408264704999, 77211968620103, 292706146651697, 1111698968597495, 4229571286335997, 16117966287887641, 61515492682026560, 235114188287816030, 899821838980825557, 3448133313264656915
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 54*x^5 + 168*x^6 + 518*x^7 + 1702*x^8 + 5672*x^9 + 19413*x^10 + 67329*x^11 + 236994*x^12 + ... such that log( A(x) ) = x + 5*x^2/2 + 13*x^3/3 + 45*x^4/4 + 131*x^5/5 + 497*x^6/6 + 1723*x^7/7 + 6525*x^8/8 + 24349*x^9/9 + 92655*x^10/10 + ... + A322203(n)*x^n/n + ... Also, A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 63*x^4 + 190*x^5 + 613*x^6 + 1976*x^7 + 6604*x^8 + 22368*x^9 + 77270*x^10 + 270208*x^11 + 956780*x^12 + ... + A322202(n)*x^n + ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1669 (first 401 terms from Paul D. Hanna)
Programs
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Maple
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end: b:= proc(n, i) option remember; `if`(n=0 or i=1, C(n), add((t-> b(t, min(t, i-1)))(n-i*j)*C(j), j=0..n/i)) end: a:= n-> b(n$2): seq(a(n), n=0..35); # Alois P. Heinz, Aug 24 2019 -
Mathematica
nmax = 25; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 12 2019 *) nmax = 25; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 12 2019 *) -
PARI
{L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );} {A322203(n) = polcoeff( n*polcoeff( L,n,x),n,y)} {a(n) = polcoeff( exp( sum(m=1,n, A322203(m)*x^m/m ) +x*O(x^n) ),n) } for(n=0,35, print1( a(n),", ") )
Formula
a(n) ~ c * 4^n / n^(3/2), where c = 1/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1))) = 0.6176761088360252844346512553859... - Vaclav Kotesovec, Jun 18 2019, updated Aug 12 2019
G.f.: Product_{j>=1} c(x^j), where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - Alois P. Heinz, Aug 24 2019
Comments