A194471
E.g.f. A(x) satisfies A(x) = exp(x) + x*A(x)^2.
Original entry on oeis.org
1, 2, 9, 79, 1065, 19401, 445933, 12389021, 403897553, 15120448273, 639345572181, 30138682861365, 1567316344601593, 89137628104427033, 5503952108613407933, 366697176991277153341, 26220726323043177903009, 2002962250253424509250081
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 79*x^3/3! + 1065*x^4/4! +...
Related expansion:
A(x)^2 = 1 + 4*x + 26*x^2/2! + 266*x^3/3! + 3880*x^4/4! + 74322*x^5/5! +...
Illustrate the recurrence:
a(2) = 1 + 2*(1*1*2 + 1*2*1) = 1 + 2*4 = 9;
a(3) = 1 + 3*(1*1*9 + 2*2*2 + 1*9*1) = 1 + 3*26 = 79;
a(4) = 1 + 4*(1*1*79 + 3*2*9 + 3*9*2 + 1*79*1) = 1 + 4*266 = 1065;
a(5) = 1 + 5*(1*1*1065 + 4*2*79 + 6*9*9 + 4*79*2 + 1*1065*1) = 1 + 5*3880 = 19401.
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f[0] = 1; f[n_] := f[n] = 1 + n*Sum[ Binomial[n - 1, k]*f[k]*f[n - 1 - k] , {k, 0, n - 1}]; Array[f, 18, 0] (* Robert G. Wilson v, Aug 25 2011 *)
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a(n):=n!*sum((k+1)^(n-k-1)*binomial(2*k,k)/(n-k)!,k,0,n); /* Vladimir Kruchinin, Sep 01 2014 */
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{a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*exp(x +O(x^(n+2))))) / (2*x),n)}
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{a(n)=1+n*sum(k=0,n-1,binomial(n-1,k)*a(k)*a(n-1-k))}
A373324
E.g.f. satisfies A(x) = exp(x) + x*A(x)^4.
Original entry on oeis.org
1, 2, 17, 349, 11249, 495401, 27715225, 1882157369, 150415131809, 13830661215649, 1438437863857961, 166962406866895817, 21396540301232809201, 3000661115664455591921, 457109095827413086174265, 75165845570197217863619161, 13270031366484750565975875905, 2503433069466253671859276038977
Offset: 0
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Table[n! * Sum[(3*k+1)^(n-k-1) * Binomial[4*k,k] / (n-k)!, {k, 0, n}], {n, 0, 20}]
A380723
E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2) / (1 - x*A(x)^2).
Original entry on oeis.org
1, 2, 21, 436, 13785, 589206, 31825381, 2080523880, 159761186577, 14097898530730, 1405926737063541, 156379679761925148, 19195200442017128425, 2577494115099820986174, 375845854490491567916805, 59145488004443221188738256, 9990898494797767848442559649, 1803160967691789114062089511250
Offset: 0
A380769
E.g.f. A(x) satisfies A(x) = exp(x / (1 - x*A(x)^2)) / (1 - x*A(x)^2).
Original entry on oeis.org
1, 2, 15, 244, 6097, 206806, 8882599, 462280960, 28279981825, 1989026203114, 158149907916031, 14028441592927180, 1373477000345414353, 147124479131269256254, 17115976784139798114775, 2149092237059821309705816, 289673905062350873773963393, 41719133895880374350508378322
Offset: 0
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a(n, q=1, r=2, s=0, t=1, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);
A381999
E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^3.
Original entry on oeis.org
1, 1, 10, 156, 3656, 115400, 4595232, 221281312, 12510826624, 812633118336, 59642105050880, 4881685773730304, 440905471531302912, 43559980305765793792, 4673231270870843441152, 541042726968231082967040, 67236501012517546330062848, 8927220151967826907452440576
Offset: 0
A377745
E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x)^2)^2.
Original entry on oeis.org
1, 3, 35, 865, 32917, 1699311, 111033607, 8788108477, 817439352233, 87406186549339, 10564550856634411, 1424421297360350169, 211968687043802337469, 34509326697582566247367, 6101526326400539736369935, 1164298084658023787974823221, 238495519792465232104337607505
Offset: 0
A377888
E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^2).
Original entry on oeis.org
1, 2, 17, 289, 7541, 267041, 11974645, 650666731, 41560476809, 3052145052433, 253400719220801, 23470964805942083, 2399562226994185885, 268404500411311273465, 32606551238103342068717, 4275233840499570086190331, 601753408713140793660643985, 90500525005651471292191270433
Offset: 0
Showing 1-7 of 7 results.
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