A377101 - OEIS

%I #21 Oct 26 2024 06:50:42

%S 1,1,2,4,12,34,108,350,1170,3980,13776,48286,171124,611994,2206018,

%T 8006120,29229932,107280390,395593940,1464883654,5445038132,

%U 20308989646,75985676050,285111556728,1072594810368,4044869862236,15287690645590,57899780087350,219708252658174,835205725190634,3180292173415490

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

%H Paul D. Hanna, <a href="/A377101/b377101.txt">Table of n, a(n) for n = 1..600</a>

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

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

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

%F (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).

%F a(n) ~ c * 4^n / n^(3/2), where c = 0.118833975923191568803966281841353477423069092... - _Vaclav Kotesovec_, Oct 26 2024

%e 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 + ...

%e where

%e 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) + ...

%e Also,

%e 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) + ...

%e SPECIFIC VALUES.

%e 3/4 = A(1/2^2) + A(1/2^3) + A(1/2^4) + A(1/2^5) + A(1/2^6) + ...

%e 4/9 = A(2/3^2) + A(2/3^3) + A(2/3^4) + A(2/3^5) + A(2/3^6) + ...

%e 5/16 = A(3/4^2) + A(3/4^3) + A(3/4^4) + A(3/4^5) + A(3/4^6) + ...

%e 6/25 = A(4/5^2) + A(4/5^3) + A(4/5^4) + A(4/5^5) + A(4/5^6) + ...

%e A(1/4) = 0.473062766764119275268472519560475021275667904444...

%e A(1/8) = 0.146091379044021979613977663433821560496912264648...

%e A(1/16) = 0.066969542527164031183514558613941039045200497375...

%e A(1/32) = 0.032291805145106464173568313939786469311507979841...

%e A(1/64) = 0.015877020134614769119124819748319680169725457847...

%e A(2/9) = 0.326160170964714161035409956881397750042473490926...

%e A(2/27) = 0.080528476310177688777131134634156506543424965067...

%e A(2/81) = 0.025332733256938154890698653999517757423697866466...

%o (PARI) \\ using formula (1)

%o {a(n) = my(V=[0,1]); for(i=1,n, V=concat(V,0); A=Ser(V);

%o 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)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) \\ using formula (2)

%o {a(n) = my(V=[0,1]); for(i=1,n, V=concat(V,0); A=Ser(V);

%o 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)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A377102, A377103, A000108.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Oct 18 2024