A171191 -id:A171191 - cp's OEIS frontend

A171199 G.f. satisfies: A(x) = exp( Sum_{n>=1} [A(x)^n + A(x)^-n]*x^n/n ).

1, 2, 3, 8, 25, 83, 289, 1041, 3847, 14504, 55569, 215727, 846761, 3354858, 13398965, 53888063, 218053915, 887107888, 3626373205, 14887942624, 61358959587, 253771944529, 1052917272543, 4381374717994, 18280470530047

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Same as A143330 after initial terms.

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 +...
log(A(x)) = [A(x)+1/A(x)]*x + [A(x)^2+1/A(x)^2]*x^2/2 + [A(x)^3+1/A(x)^3]*x^3/3 +...

Cf. A171190, A171191, A143330.

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,(A^m+A^-m+x*O(x^n))*x^m/m)));polcoeff(A,n)}

{a(n)=polcoeff((1+x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x), n)}

G.f.: A(x) = (1+x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).
G.f. satisfies: 1 = (A(x) - x)*(1 - x*A(x)).
a(0) = 1, a(1) = 2; a(n) = a(n-1) + a(n-2) + Sum_{k=2..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 20 2021

A171190 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).

Original entry on oeis.org

1, 2, 3, 10, 27, 112, 336, 1490, 4791, 22138, 74079, 351288, 1207620, 5831208, 20436516, 100004994, 355610367, 1758044950, 6322608561, 31511387450, 114359284515, 573713781760, 2097612975456, 10580600244664, 38925304968612

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 10*x^3 + 27*x^4 + 112*x^5 + 336*x^6 + 1490*x^7 + 4791*x^8 + 22138*x^9 + 74079*x^10 + 351288*x^11 + 1207620*x^12 + ...
where the logarithm of A(x) may be written as
log(A(x)) = (A(x) + A(-x))*x + (A(x)^2 + A(-x)^2)*x^2/2 + (A(x)^3 + A(-x)^3)*x^3/3 + (A(x)^4 + A(-x)^4)*x^4/4 + (A(x)^5 + A(-x)^5)*x^5/5 + ...
Incidentally, the square root of g.f. A(x) is an integer series starting
A(x)^(1/2) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 43*x^5 + 108*x^6 + 558*x^7 + 1517*x^8 + 8175*x^9 + 23219*x^10 + 128516*x^11 + 375896*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A171191, A171199.

{a(n) = my(A=1+x+x*O(x^n)); for(i=1,n, A=exp(sum(m=1,n,(A^m+subst(A^m,x,-x)+x*O(x^n))*x^m/m)));polcoeff(A,n)}

{a(n) = my(A=1+x); for(i=1,n, A=(1-x*A+x*O(x^n))^-1*(1-x*subst(A,x,-x)+x*O(x^n))^-1);polcoeff(A,n)} \\ Paul D. Hanna, Dec 06 2009

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).
(2) A(x) = 1/((1 - x*A(x)) * (1 - x*A(-x))). - Paul D. Hanna, Dec 06 2009
(3) 0 = 1 - (3-x)*A(x) + (2-x)*A(x)^2 - (2-5*x)*x*A(x)^3 - (2+x)*x^2*A(x)^4 + 2*x^3*A(x)^5. - Paul D. Hanna, Feb 11 2024

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A171199 G.f. satisfies: A(x) = exp( Sum_{n>=1} [A(x)^n + A(x)^-n]*x^n/n ).

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A171190 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x)^n + A(-x)^n) * x^n/n ).

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