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A200075 G.f. satisfies A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)^3).

Original entry on oeis.org

1, 1, 3, 11, 45, 198, 914, 4367, 21414, 107155, 544987, 2808978, 14640073, 77025373, 408544815, 2182206259, 11727989593, 63373962690, 344109933186, 1876562458845, 10273572074493, 56443282489240, 311097732946200, 1719707775782826, 9531914043637385, 52963938340248863, 294966593345731623
Offset: 0

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Author

Paul D. Hanna, Nov 13 2011

Keywords

Comments

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 198*x^5 + 914*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 121*x^4 + 552*x^5 + 2615*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 52*x^3 + 237*x^4 + 1122*x^5 + 5463*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 125*x^3 + 630*x^4 + 3211*x^5 + 16545*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^5.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A)*x*A + (1 + 2^2*x*A + x^2*A^2)*x^2*A^2/2 +
(1 + 3^2*x*A + 3^2*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +
(1 + 4^2*x*A + 6^2*x^2*A^2 + 4^2*x^3*A^3 + x^4*A^4)*x^4*A^4/4 +
(1 + 5^2*x*A + 10^2*x^2*A^2 + 10^2*x^3*A^3 + 5^2*x^4*A^4 + x^5*A^5)*x^5*A^5/5 +
(1 + 6^2*x*A + 15^2*x^2*A^2 + 20^2*x^3*A^3 + 15^2*x^4*A^4 + 6^2*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 129*x^4/4 + 686*x^5/5 + 3713*x^6/6 + 20350*x^7/7 +...
Given G(x) where 1+x*G(x) is the g.f. of A004148, then the coefficients in the powers of G(x) begin:
1: [(1), 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, ...];
2: [1,(2), 5, 12, 28, 66, 156, 370, 882, 2112, ...];
3: [1, 3,(9), 25, 66, 171, 437, 1107, 2790, 7009, ...];
4: [1, 4, 14,(44), 129, 364, 1000, 2696, 7172, 18892, ...];
5: [1, 5, 20, 70,(225), 686, 2015, 5760, 16135, 44500, ...];
6: [1, 6, 27, 104, 363,(1188), 3713, 11214, 32994, 95106, ...];
7: [1, 7, 35, 147, 553, 1932,(6398), 20350, 62734, 188650, ...];
8: [1, 8, 44, 200, 806, 2992, 10460,(34936), 112585, 352560, ...];
9: [1, 9, 54, 264, 1134, 4455, 16389, 57330,(192726), 627406, ...]; ...;
the coefficients in parenthesis form the initial terms of this sequence:
[1/1, 2/2, 9/3, 44/4, 225/5, 1188/6, 6398/7, 34936/8, 192726/9, ...].
The coefficients in the logarithm of the g.f. is also a diagonal in the above table.

Links

Crossrefs

Cf. A004148, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.
Cf. A186241, A199874, A200074, A200718, A200719, A215576.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x*(1-x-x^2 + Sqrt[(1+x+x^2)*(1-3*x+x^2)])/2,{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 19 2013 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1+x*A^2)*(1+x^2*A^3)+x*O(x^n));polcoeff(A,n)}
  • PARI
    {a(n)=polcoeff(1/x*serreverse(x*(1-x-x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)+x*O(x^n)))/2),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^j*A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x*A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*A^j)*x^m*A^m/m))); polcoeff(A, n, x)}

Formula

G.f.: (1/x)*Series_Reversion( x*(1-x-x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)))/2 ).
a(n) = [x^n] G(x)^(n+1)/(n+1), where 1+x*G(x) is the g.f. of A004148.
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x/G(x) ) where 1+x*G(x) is the g.f. of A004148.
(2) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and 1+x*G(x) is the g.f. of A004148.
(3) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^k) * x^n*A(x)^n/n ).
(4) A(x) = exp( Sum_{n>=1} (1-x*A(x))^(2*n+1)*(Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^k) * x^n*A(x)^n/n ).
Recurrence: 8*n*(2*n+1)*(4*n+1)*(4*n+3)*(1557671*n^7 - 18939961*n^6 + 94817789*n^5 - 252067387*n^4 + 381880748*n^3 - 327052012*n^2 + 145198992*n - 25583040)*a(n) = (2026529971*n^11 - 24640889261*n^10 + 122927623620*n^9 - 322351865586*n^8 + 467303512311*n^7 - 343677276405*n^6 + 61590777290*n^5 + 76066203476*n^4 - 45605627832*n^3 + 4625651136*n^2 + 1916801280*n - 338688000)*a(n-1) + 2*(800642894*n^11 - 10936104295*n^10 + 62803409541*n^9 - 196202081616*n^8 + 357730085364*n^7 - 370711524567*n^6 + 174415015309*n^5 + 25877389846*n^4 - 63266190708*n^3 + 19055552472*n^2 + 1313789760*n - 861840000)*a(n-2) + 6*(308418858*n^11 - 4675368852*n^10 + 30103912361*n^9 - 106665982366*n^8 + 223860428776*n^7 - 274000455628*n^6 + 166116940489*n^5 - 2432493994*n^4 - 54297743044*n^3 + 22033617000*n^2 + 936446400*n - 1315440000)*a(n-3) + 6*(n-2)*(2*n-7)*(3*n-10)*(3*n-8)*(1557671*n^7 - 8036264*n^6 + 13889114*n^5 - 7559372*n^4 - 2491645*n^3 + 2975476*n^2 - 179460*n - 187200)*a(n-4). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 1301/1024 + 1/(1024*sqrt(3/(7183147 - (2002819072*2^(2/3))/(3725055779 + 42057117*sqrt(16305))^(1/3) + 1024*(7450111558 + 84114234*sqrt(16305))^(1/3)))) + (1/2)*sqrt(7183147/393216 - (3725055779 + 42057117*sqrt(16305))^(1/3)/(384*2^(2/3)) + 977939/(192*(7450111558 + 84114234*sqrt(16305))^(1/3)) + (1/131072)*(4194454317*sqrt(3/(7183147 - (2002819072*2^(2/3))/(3725055779 + 42057117*sqrt(16305))^(1/3) + 1024*(7450111558 + 84114234*sqrt(16305))^(1/3))))) = 5.89828930084513611... is the root of the equation -108 - 1188*d - 1028*d^2 - 1301*d^3 + 256*d^4 = 0 and c = 0.656947859044624009263362998790812821830934... - Vaclav Kotesovec, Sep 19 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-k+1,k) * binomial(2*n-k+1,n-2*k) / (2*n-k+1). - Seiichi Manyama, Jul 18 2023

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