A377101 G.f. A(x) satisfies Sum_{n>=1} A(x^n - x^(n+1)) = x + x^2.

1, 1, 2, 4, 12, 34, 108, 350, 1170, 3980, 13776, 48286, 171124, 611994, 2206018, 8006120, 29229932, 107280390, 395593940, 1464883654, 5445038132, 20308989646, 75985676050, 285111556728, 1072594810368, 4044869862236, 15287690645590, 57899780087350, 219708252658174, 835205725190634, 3180292173415490

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..600

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) x + x^2 = Sum_{n>=1} A(x^n - x^(n+1)).

(2) x + x^2 = Sum_{n>=1} a(n) * x^n * (1-x)^n/(1-x^n).

(3) 2*C(x) - x = Sum_{n>=1} a(n) * x^n / (1 - C(x)^n), where C(x) = (1 - sqrt(1 - 4*x))/2 is the g.f. of the Catalan numbers (A000108).

a(n) ~ c * 4^n / n^(3/2), where c = 0.118833975923191568803966281841353477423069092... - Vaclav Kotesovec, Oct 26 2024

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 34*x^6 + 108*x^7 + 350*x^8 + 1170*x^9 + 3980*x^10 + 13776*x^11 + 48286*x^12 + ...
where
x + x^2 = A(x-x^2) + A(x^2-x^3) + A(x^3-x^4) + A(x^4-x^5) + A(x^5-x^6) + ...
Also,
x + x^2 = a(1)*x*(1-x)/(1-x) + a(2)*x^2*(1-x)^2/(1-x^2) + a(3)*x^3*(1-x)^3/(1-x^3) + a(4)*x^4*(1-x)^4/(1-x^4) + ...
SPECIFIC VALUES.
3/4 = A(1/2^2) + A(1/2^3) + A(1/2^4) + A(1/2^5) + A(1/2^6) + ...
4/9 = A(2/3^2) + A(2/3^3) + A(2/3^4) + A(2/3^5) + A(2/3^6) + ...
5/16 = A(3/4^2) + A(3/4^3) + A(3/4^4) + A(3/4^5) + A(3/4^6) + ...
6/25 = A(4/5^2) + A(4/5^3) + A(4/5^4) + A(4/5^5) + A(4/5^6) + ...
A(1/4) = 0.473062766764119275268472519560475021275667904444...
A(1/8) = 0.146091379044021979613977663433821560496912264648...
A(1/16) = 0.066969542527164031183514558613941039045200497375...
A(1/32) = 0.032291805145106464173568313939786469311507979841...
A(1/64) = 0.015877020134614769119124819748319680169725457847...
A(2/9) = 0.326160170964714161035409956881397750042473490926...
A(2/27) = 0.080528476310177688777131134634156506543424965067...
A(2/81) = 0.025332733256938154890698653999517757423697866466...

Prog

(PARI) \\ using formula (1)
{a(n) = my(V=[0, 1]); for(i=1, n, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef(x+x^2 - sum(m=1, #V, subst(A, x, x^m*(1 - x) +x*O(x^#V))), #V-1) ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) \\ using formula (2)
{a(n) = my(V=[0, 1]); for(i=1, n, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef(x+x^2 - sum(m=1, #V-1, V[m+1]*x^m*(1-x)^m/(1-x^m +O(x^#V))), #V-1) ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A377102, A377103, A000108.

Sequence in context: A148198 A148199 A343663 * A108530 A001895 A267618

Adjacent sequences: A377098 A377099 A377100 * A377102 A377103 A377104

Keyword

nonn

Author

Paul D. Hanna, Oct 18 2024

Status

approved