A218223
G.f. A(x) satisfies: A(x) = 1+x + x^2*[d/dx A(x)^3].
Original entry on oeis.org
1, 1, 3, 24, 273, 3996, 70785, 1465506, 34662222, 921511944, 27201024639, 882828325530, 31253560065684, 1198758613494852, 49530067909218819, 2193498057583259784, 103664556373964098860, 5207896547115772335552, 277161367378578537506868
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 273*x^4 + 3996*x^5 + 70785*x^6 +...
Related series:
A(x)^3 = 1 + 3*x + 12*x^2 + 91*x^3 + 999*x^4 + 14157*x^5 + 244251*x^6 +...
d/dx A(x)^3 = 3 + 24*x + 273*x^2 + 3996*x^3 + 70785*x^4 + 1465506*x^5 +...
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{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^3+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A218224
G.f. A(x) satisfies: A(x) = 1+x + x^2*[d/dx A(x)^4].
Original entry on oeis.org
1, 1, 4, 44, 684, 13636, 328000, 9198240, 294075040, 10549834368, 419626384128, 18330935118080, 872618259925632, 44970631837229184, 2494887017741434368, 148272655438005392896, 9399158287979230003200, 633107847492164526284800, 45159576693655485274008576
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 44*x^3 + 684*x^4 + 13636*x^5 + 328000*x^6 +...
Related series:
A(x)^3 = 1 + 4*x + 22*x^2 + 228*x^3 + 3409*x^4 + 65600*x^5 + 1533040*x^6 +...
d/dx A(x)^3 = 4 + 44*x + 684*x^2 + 13636*x^3 + 328000*x^4 + 9198240*x^5 +...
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{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^4+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A386208
G.f. A(x) satisfies A(x) = 1/(1-x) + x^2 * (d/dx A(x)^2).
Original entry on oeis.org
1, 1, 3, 15, 109, 1029, 11831, 159595, 2466073, 42920585, 830791243, 17706459431, 412116616517, 10403094478669, 283137307529727, 8266131486719107, 257710382446835761, 8546074646120275473, 300384437888406796051, 11155675460369469443263, 436506923733804200244509
Offset: 0
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+(i-1)*sum(j=0, i-1, v[j+1]*v[i-j])); v;
A218222
G.f. A(x) satisfies: A(x) = x + x*[d/dx A(x)^2].
Original entry on oeis.org
1, 2, 12, 112, 1360, 19872, 335104, 6359040, 133560576, 3069007360, 76493880320, 2054400577536, 59136549994496, 1816392567062528, 59305340822814720, 2051451257317490688, 74958908119819812864, 2885480280276224311296, 116731741304854533111808
Offset: 1
G.f.: A(x) = x + 2*x^2 + 12*x^3 + 112*x^4 + 1360*x^5 + 19872*x^6 +...
Related series:
A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 272*x^5 + 3312*x^6 + 47872*x^7 + 794880*x^8 + 14840064*x^9 +...+ A112915(n-1)*x^n +...
d/dx A(x)^2 = 2*x + 12*x^2 + 112*x^3 + 1360*x^4 + 19872*x^5 +...
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a:= proc(n) option remember; `if`(n<2, 1,
n*add(a(i)*a(n-i), i=1..n-1))
end:
seq(a(n), n=1..20); # Alois P. Heinz, Nov 05 2020
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a[n_] := a[n] = If[n<2, 1, n*Sum[a[i]*a[n-i], {i, 1, n-1}]];
Array[a, 20] (* Jean-François Alcover, Dec 18 2020, after Maple *)
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{a(n)=local(A=x+x^2); for(i=1, n, A=x+x*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A354737
a(0) = a(1) = 1; a(n) = n * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 2, 6, 20, 80, 336, 1568, 7584, 39312, 210080, 1180256, 6813312, 40890304, 251528704, 1597332480, 10376040448, 69259146752, 472084038144, 3295588345344, 23459477468160, 170610216311808, 1263629972183040, 9543419750909952, 73322350509367296, 573544008429363200
Offset: 0
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a[0] = a[1] = 1; a[n_] := a[n] = n Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + x + 2 x^2 A[x]^2 + 2 x^3 A[x] D[A[x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A354738
a(0) = a(1) = 1; a(n) = (n-1) * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 1, 4, 9, 40, 135, 636, 2688, 13552, 65871, 355520, 1906740, 10963656, 63468171, 386532944, 2383820820, 15294890848, 99626199832, 670333562352, 4583302104450, 32213942456000, 230118463761795, 1683896120829384, 12520330728001670, 95110075114630416
Offset: 0
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a[0] = a[1] = 1; a[n_] := a[n] = (n - 1) Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + x + x^2 A[x]^2 + 2 x^3 A[x] D[A[x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Showing 1-6 of 6 results.