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A200028 G.f. satisfies: A(x) = 1 + x*A(x)^3 + 2*x^2*A(x)^2 + x^3*A(x). 3
1, 1, 5, 23, 123, 714, 4371, 27789, 181703, 1214166, 8254642, 56915989, 397055129, 2797402939, 19875881883, 142255845279, 1024672816559, 7422364394376, 54034027247172, 395121014784661, 2900916414217452, 21375452831552800, 158025136138734047, 1171770189246765570 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

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

Recurrence: 2*n*(2*n+1)*(269*n^4 - 2870*n^3 + 11215*n^2 - 17566*n + 7440)*a(n) = 3*(2421*n^6 - 28251*n^5 + 127807*n^4 - 266413*n^3 + 251420*n^2 - 110096*n + 18072)*a(n-1) + 18*(538*n^6 - 7085*n^5 + 36497*n^4 - 91805*n^3 + 108177*n^2 - 38330*n - 12672)*a(n-2) + 2*(n-4)*(1076*n^5 - 11480*n^4 + 46507*n^3 - 83710*n^2 + 43773*n + 29106)*a(n-3) + 36*(n-5)*(2*n-3)*(7*n^2 - 3*n + 38)*a(n-4) - 4*(n-6)*(269*n^5 - 3139*n^4 + 13659*n^3 - 25097*n^2 + 16486*n - 4230)*a(n-5) - 2*(n-7)*(2*n-11)*(269*n^4 - 1794*n^3 + 4219*n^2 - 2670*n - 1512)*a(n-6). - Vaclav Kotesovec, Sep 10 2013

a(n) ~ c*d^n/n^(3/2), where d = 7.918229914943462873... is the root of the equation 4 + 4*d - 8*d^3 - 36*d^4 - 27*d^5 + 4*d^6 = 0 and c = 0.283739249... - Vaclav Kotesovec, Sep 10 2013

EXAMPLE

G.f.: A(x) = 1 + x + 5*x^2 + 23*x^3 + 123*x^4 + 714*x^5 + 4371*x^6 +...

Related expansions:

A(x)^2 = 1 + 2*x + 11*x^2 + 56*x^3 + 317*x^4 + 1904*x^5 + 11929*x^6 +...

A(x)^3 = 1 + 3*x + 18*x^2 + 100*x^3 + 597*x^4 + 3714*x^5 + 23858*x^6 +...

MATHEMATICA

nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[1+x*AGF*(AGF+x)^2-AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 10 2013 *)

PROG

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

CROSSREFS

Cf. A200029, A200030.

Sequence in context: A151881 A229811 A121636 * A020032 A293088 A186755

Adjacent sequences:  A200025 A200026 A200027 * A200029 A200030 A200031

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 12 2011

STATUS

approved

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Last modified August 20 10:05 EDT 2019. Contains 326149 sequences. (Running on oeis4.)