A200075 - OEIS

%I #27 Jul 18 2023 10:32:50

%S 1,1,3,11,45,198,914,4367,21414,107155,544987,2808978,14640073,

%T 77025373,408544815,2182206259,11727989593,63373962690,344109933186,

%U 1876562458845,10273572074493,56443282489240,311097732946200,1719707775782826,9531914043637385,52963938340248863,294966593345731623

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

%C More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:

%C 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)] ),

%C then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).

%H G. C. Greubel, <a href="/A200075/b200075.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1/x)*Series_Reversion( x*(1-x-x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)))/2 ).

%F a(n) = [x^n] G(x)^(n+1)/(n+1), where 1+x*G(x) is the g.f. of A004148.

%F G.f. A(x) satisfies:

%F (1) A(x) = (1/x)*Series_Reversion( x/G(x) ) where 1+x*G(x) is the g.f. of A004148.

%F (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.

%F (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 ).

%F (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.

%F 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

%F 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

%F 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

%e G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 198*x^5 + 914*x^6 +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 121*x^4 + 552*x^5 + 2615*x^6 +...

%e A(x)^3 = 1 + 3*x + 12*x^2 + 52*x^3 + 237*x^4 + 1122*x^5 + 5463*x^6 +...

%e A(x)^5 = 1 + 5*x + 25*x^2 + 125*x^3 + 630*x^4 + 3211*x^5 + 16545*x^6 +...

%e where A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^5.

%e The logarithm of the g.f. A = A(x) equals the series:

%e log(A(x)) = (1 + x*A)*x*A + (1 + 2^2*x*A + x^2*A^2)*x^2*A^2/2 +

%e (1 + 3^2*x*A + 3^2*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +

%e (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 +

%e (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 +

%e (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 +...

%e more explicitly,

%e 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 +...

%e Given G(x) where 1+x*G(x) is the g.f. of A004148, then the coefficients in the powers of G(x) begin:

%e 1: [(1), 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, ...];

%e 2: [1,(2), 5, 12, 28, 66, 156, 370, 882, 2112, ...];

%e 3: [1, 3,(9), 25, 66, 171, 437, 1107, 2790, 7009, ...];

%e 4: [1, 4, 14,(44), 129, 364, 1000, 2696, 7172, 18892, ...];

%e 5: [1, 5, 20, 70,(225), 686, 2015, 5760, 16135, 44500, ...];

%e 6: [1, 6, 27, 104, 363,(1188), 3713, 11214, 32994, 95106, ...];

%e 7: [1, 7, 35, 147, 553, 1932,(6398), 20350, 62734, 188650, ...];

%e 8: [1, 8, 44, 200, 806, 2992, 10460,(34936), 112585, 352560, ...];

%e 9: [1, 9, 54, 264, 1134, 4455, 16389, 57330,(192726), 627406, ...]; ...;

%e the coefficients in parenthesis form the initial terms of this sequence:

%e [1/1, 2/2, 9/3, 44/4, 225/5, 1188/6, 6398/7, 34936/8, 192726/9, ...].

%e The coefficients in the logarithm of the g.f. is also a diagonal in the above table.

%t 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 *)

%o (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)}

%o (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)}

%o (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)}

%o (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)}

%Y Cf. A004148, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

%Y Cf. A186241, A199874, A200074, A200718, A200719, A215576.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 13 2011