The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A182263 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x)^2. 2
 1, 1, 6, 91, 2910, 187178, 24019884, 6154080275, 3151538898870, 3227331249742334, 6609648919647088788, 27073195436180090799006, 221783764770326660974008300, 3633705802215756626623500731892, 119069276624759801067298501607804376 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..70 FORMULA a(n) = (2^n-1) * { [x^(n-1)] A(x)^2 } for n>0 with a(0)=1. a(n) = (2^n-1) * Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1. a(n) ~ c * 2^((n-1)*(n+4)/2), where c = 0.71662215139236633556752111264619992099204134882... - Vaclav Kotesovec, Feb 22 2014 EXAMPLE G.f.: A(x) = 1 + x + 6*x^2 + 91*x^3 + 2910*x^4 + 187178*x^5 + 24019884*x^6 +... Related expansions: A(x)^2 = 1 + 2*x + 13*x^2 + 194*x^3 + 6038*x^4 + 381268*x^5 + 48457325*x^6 + 12358976074*x^7 + 6315716731394*x^8 + 6461044887240556*x^9 +... such that a(n) = (2^n-1) times the coefficient of x^(n-1) in A(x)^2: a(2) = 3 * 2 = 6; a(3) = 7 * 13 = 91; a(4) = 15 * 194 = 2910; a(5) = 31 * 6038 = 187178; a(6) = 63 * 381268 = 24019884; ... MATHEMATICA a = ConstantArray[0, 21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = (2^n-1)* Sum[a[[k+1]]*a[[n-k]], {k, 0, n-1}], {n, 2, 20}]; a (* Vaclav Kotesovec, Feb 22 2014 *) PROG (PARI) /* Generating Function Satisfies: */ {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} /* = n-th derivative of F */ {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(k=1, n, x^k/k!*Dx(k, x*A^2+x*O(x^n) ))); polcoeff(A, n)} (PARI) /* Recurrence: */ {a(n)=if(n==0, 1, (2^n-1)*sum(k=0, n-1, a(k)*a(n-k-1)))} for(n=0, 15, print1(a(n), ", ")) (PARI) /* Recurrence: */ {a(n)=local(A=1+sum(k=1, n-1, a(k)*x^k)+x*O(x^n)); if(n==0, 1, (2^n-1)*polcoeff(A^2, n-1))} CROSSREFS Cf. A005329, A182264. Sequence in context: A219220 A006151 A005327 * A171910 A278683 A280214 Adjacent sequences: A182260 A182261 A182262 * A182264 A182265 A182266 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 21 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)