A200719 - OEIS

%I #20 Jul 18 2023 12:26:17

%S 1,1,3,13,64,340,1903,11053,65993,402527,2497439,15712220,100001459,

%T 642719263,4165537744,27193644061,178654643151,1180282875483,

%U 7836312619243,52259258911091,349902441457427,2351240866736891,15851508780927739,107187240225220684,726784821098903319

%N G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x^2*A(x)^5).

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

%C F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),

%C then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).

%H Seiichi Manyama, <a href="/A200719/b200719.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vaclav Kotesovec)

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

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

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

%F Recurrence: 4232*(n-2)*(n-1)*n*(2*n - 3)*(2*n - 1)*(2*n + 1)*(108983978975*n^7 - 1828734495225*n^6 + 13017379495661*n^5 - 50928975062019*n^4 + 118201965098732*n^3 - 162617590602876*n^2 + 122676758610192*n - 39103265134080)*a(n) = 8*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(850837923857825*n^9 - 15127768128079400*n^8 + 116088908648008427*n^7 - 502364025369222635*n^6 + 1342887860190877280*n^5 - 2280899268898038065*n^4 + 2433907848768834828*n^3 - 1548429898790214180*n^2 + 521138603722292640*n - 68863424146977600)*a(n-1) - 30*(n-2)*(2*n - 3)*(155302170039375*n^11 - 3227155335853125*n^10 + 29807524885054600*n^9 - 161278340404759950*n^8 + 566950865855228019*n^7 - 1356848300481904461*n^6 + 2250482361655315470*n^5 - 2579665279074165840*n^4 + 1996011605601581864*n^3 - 988803599084885136*n^2 + 280851990522009984*n - 34444332223983360)*a(n-2) + 5*(7288303593953125*n^13 - 187891351713750000*n^12 + 2204843674914291875*n^11 - 15579013461781304250*n^10 + 73867718896175411475*n^9 - 247858726321141236540*n^8 + 604530296941440837821*n^7 - 1082990060568950070282*n^6 + 1421457900098213642392*n^5 - 1345695224728829837040*n^4 + 889319601933492222864*n^3 - 386196670582228097568*n^2 + 97916706472751405568*n - 10797892365692920320)*a(n-3) + 10*(n-3)*(2*n - 7)*(5*n - 18)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(108983978975*n^7 - 1065846642400*n^6 + 4333636082786*n^5 - 9458655747964*n^4 + 11899609166891*n^3 - 8554104592084*n^2 + 3208950812340*n - 473478110640)*a(n-4). - _Vaclav Kotesovec_, Nov 17 2017

%F a(n) ~ s * sqrt((r*s*(r*s^3 - 1) - 3) / (7*Pi*(5*r*s*(1 + r*s^3) - 3))) / (2*n^(3/2)*r^n), where r = 0.1385102270697349252376651829944449360743895474888... and s = 1.450646440303399446510765649245639306003224666768... are real roots of the system of equations (1 + r*s^2)*(1 + r^2*s^5) = s, r*s*(2 + 5*r*s^3 + 7*r^2*s^5) = 1. - _Vaclav Kotesovec_, Nov 22 2017

%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 + 13*x^3 + 64*x^4 + 340*x^5 + 1903*x^6 +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 163*x^4 + 886*x^5 + 5039*x^6 +...

%e A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 765*x^4 + 4481*x^5 + 26920*x^6 +...

%e A(x)^7 = 1 + 7*x + 42*x^2 + 252*x^3 + 1533*x^4 + 9457*x^5 + 59101*x^6 +...

%e where A(x) = 1 + x*A(x)^2 + x^2*A(x)^5 + x^3*A(x)^7.

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

%e log(A(x)) = (1 + x*A^3)*x*A + (1 + 2^2*x*A^3 + x^2*A^6)*x^2*A^2/2 +

%e (1 + 3^2*x*A^3 + 3^2*x^2*A^6 + x^3*A^9)*x^3*A^3/3 +

%e (1 + 4^2*x*A^3 + 6^2*x^2*A^6 + 4^2*x^3*A^9 + x^4*A^12)*x^4*A^4/4 +

%e (1 + 5^2*x*A^3 + 10^2*x^2*A^6 + 10^2*x^3*A^9 + 5^2*x^4*A^12 + x^5*A^15)*x^5*A^5/5 + ...

%o (PARI) {a(n)=local(p=1,q=3,A=1+x);for(i=1,n,A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n));polcoeff(A,n)}

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

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

%Y Cf. A200716, A200717, A200718, A200074, A200075, A199876, A199874, A199877, A198951, A198953, A198957, A192415, A198888, A036765, A104545.

%Y Cf. A186241, A199874, A200074, A200075, A200718, A215576.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 21 2011