A198888 G.f. A(x) satisfies A(x) = (1 + x*A(x))*(1 + x^3*A(x)^4).

1, 1, 1, 2, 7, 22, 61, 172, 528, 1695, 5447, 17486, 56778, 187064, 622149, 2080325, 6990670, 23621143, 80230388, 273687898, 937072049, 3219316096, 11095261035, 38351414036, 132915860364, 461770505371, 1607875309626, 5610314558562, 19614016834508, 68696001390320, 241007011551493

Offset: 0

Author

Paul D. Hanna, Nov 03 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 61*x^6 + 172*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 71*x^4 + 236*x^5 + 766*x^6 +...
A(x)^5 = 1 + 5*x + 15*x^2 + 40*x^3 + 120*x^4 + 401*x^5 + 1340*x^6 +...
where A(x) = 1 + x*A(x) + x^3*A(x)^4 + x^4*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2*A(x)^3)*x + (1 + 2^2*x^2*A(x)^3 + x^4*A(x)^6)*x^2/2 +
(1 + 3^2*x^2*A(x)^3 + 3^2*x^4*A(x)^6 + x^6*A(x)^9)*x^3/3 +
(1 + 4^2*x^2*A(x)^3 + 6^2*x^4*A(x)^6 + 4^2*x^6*A(x)^9 + x^8*A(x)^12)*x^4/4 +
(1 + 5^2*x^2*A(x)^3 + 10^2*x^4*A(x)^6 + 10^2*x^6*A(x)^9 + 5^2*x^8*A(x)^12 + x^10*A(x)^15)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 76*x^5/5 + 232*x^6/6 + 743*x^7/7 + 2629*x^8/8 + 9481*x^9/9 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A181734, A198957, A198953, A198951, A192415, A036765.

Programs

Mathematica

Table[Sum[Binomial[n+k,k]*Binomial[n+k+1,n-3*k]/(n+1),{k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

PARI

{a(n)=sum(k=0, n\3, binomial(n+k, k)*binomial(n+k+1, n-3*k))/(n+1)}

PARI

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^3*(A+x*O(x^n))^4)); polcoeff(A, n)}

PARI

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(x^2*A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

PARI

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, n, binomial(m+j, j)^2*(x^2*A^3+x*O(x^n))^j)*(1-x^2*A^3)^(2*m+1)*x^m/m))); polcoeff(A, n, x)}

Formula

a(n) = Sum_{k=0..[n/3]} C(n+k, k)*C(n+k+1, n-3*k)/(n+1).

G.f. A(x) satisfies:

(1) A(x) = (1/x)*Series_Reversion( x/(1+x) - x^4 ). [Corrected by Seiichi Manyama, Dec 15 2024]

(2) A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 + x)/(1 - x^3 - x^4).

(3) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*x^(2*k)*A(x)^(3*k)] * x^n/n ).

(4) A(x) = exp( Sum_{n>=1} [Sum_{k>=0} C(n+k,k)^2*x^(2*k)*A(x)^(3*k)]*(1-x^2*A(x)^3)^(2*n+1)* x^n/n ).

Recurrence: 283*(n-2)*(n-1)*n*(n+1)*(23959952*n^4 - 257205740*n^3 + 1013304652*n^2 - 1735060589*n + 1087154052)*a(n) = 4*(n-2)*(n-1)*n*(8529742912*n^5 - 95830114896*n^4 + 406564828744*n^3 - 799079033082*n^2 + 700270562579*n - 198783157747)*a(n-1) - 8*(n-2)*(n-1)*(8625582720*n^6 - 109845231840*n^5 + 557377471920*n^4 - 1435513153260*n^3 + 1966313576808*n^2 - 1346689501571*n + 355664911636)*a(n-2) + 32*(n-2)*(4216951552*n^7 - 64244492224*n^6 + 407865945256*n^5 - 1396107234938*n^4 + 2774470392903*n^3 - 3187035309382*n^2 + 1946241786026*n - 482103205479)*a(n-3) - 16*(n-2)*(1150077696*n^7 - 19246341696*n^6 + 133834520688*n^5 - 499899483140*n^4 + 1078973257808*n^3 - 1338172075263*n^2 + 875535465587*n - 229801752572)*a(n-4) + 8*(n-4)*(2*n - 5)*(4*n - 17)*(4*n - 11)*(23959952*n^4 - 161365932*n^3 + 385447144*n^2 - 384228697*n + 132152327)*a(n-5). - Vaclav Kotesovec, Sep 18 2013

a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3.686367878047643633... is the root of the equation -256 + 768*d - 5632*d^2 + 2880*d^3 - 1424*d^4 + 283*d^5 = 0 and c = 0.73361916425726935915879240304621641469885... - Vaclav Kotesovec, Sep 18 2013