A322202 G.f.: exp( Sum_{n>=1} A322201(n)*x^n/n ), where A322201(n) is the coefficient of x^n*y^n/(2*n) in Sum_{n>=1} -log(1 - (x^n + y^n)).

1, 2, 7, 20, 63, 190, 613, 1976, 6604, 22368, 77270, 270208, 956780, 3419212, 12323226, 44723840, 163320766, 599601984, 2211844684, 8193734760, 30469278673, 113692852342, 425558528235, 1597428832560, 6011972255226, 22680620270712, 85754229105470, 324898592591960, 1233299357981416, 4689870496585016, 17863799895741982, 68149300647823612, 260364494604701847, 996086232267182566, 3815683108118138847, 14634441964549504036

Offset: 0

Author

Paul D. Hanna, Nov 30 2018

Keywords

• nonn

Comments

Conjecture: Euler transform of A123611. - Vaclav Kotesovec, Dec 12 2020

Examples

			G.f.: A(x) = 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 + ...
such that
log( A(x) ) = 2*x + 10*x^2/2 + 26*x^3/3 + 90*x^4/4 + 262*x^5/5 + 994*x^6/6 + 3446*x^7/7 + 13050*x^8/8 + 48698*x^9/9 + 185310*x^10/10 + ... + A322201(n)*x^n/n + ...
sqrt(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 + ... + A322204(n)*x^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A322200, A322201, A322204.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*x^(j*k), {k, 0, nmax/j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 12 2020 *)
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*x^k])/(2*x^k), {k, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 12 2020 *)

PARI

{L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{A322201(n) = polcoeff( 2*n*polcoeff( L,n,x),n,y)}
{a(n) = polcoeff( exp( sum(m=1,n, A322201(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 = 4/sqrt(Pi) * Product_{j>=1} (2^(j+1) * (2^j - sqrt(4^j - 1)))^2 = 2.704933139869066452954644773467... - Vaclav Kotesovec, Jun 18 2019, updated Dec 12 2020

G.f.: Product_{j>=1} c(x^j)^2, where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108. - Vaclav Kotesovec, Dec 12 2020