A185183
G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].
Original entry on oeis.org
1, 1, 2, 10, 72, 672, 7640, 102072, 1564864, 27064448, 521248320, 11064781760, 256702399360, 6462978471168, 175520877380992, 5115062135795584, 159227683153536000, 5273353734210310144, 185143079148664099840, 6869062513111759635456
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 72*x^4 + 672*x^5 + 7640*x^6 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 24*x^3 + 168*x^4 + 1528*x^5 + 17012*x^6 +...
d/dx A(x)^2 = 2 + 10*x + 72*x^2 + 672*x^3 + 7640*x^4 + 102072*x^5 +...
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{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^2+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;
A386263
G.f. A(x) satisfies A(x) = 1/( (1-x)^2 * (1 - x*A(x) - 2*x^2*A'(x)) ).
Original entry on oeis.org
1, 3, 15, 121, 1333, 18091, 286867, 5158385, 103226313, 2269474723, 54307112951, 1404350909545, 39020894189245, 1159475912653163, 36695329075865083, 1232560854783934561, 43801551907551784721, 1642199848241650875907, 64789265823476378293855
Offset: 0
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terms=19; A[]=1; Do[A[x]=1/( (1-x)^2 * (1 - x*A[x] - 2*x^2*A'[x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 17 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=i+1+i*sum(j=0, i-1, v[j+1]*v[i-j])); v;
A386264
G.f. A(x) satisfies A(x) = 1/( (1-x)^3 * (1 - x*A(x) - 2*x^2*A'(x)) ).
Original entry on oeis.org
1, 4, 22, 190, 2239, 32431, 546832, 10414132, 219845677, 5079617326, 127292440606, 3435881715553, 99351006910147, 3063829146597493, 100385767850729656, 3483114440490487576, 127610457725933245753, 4923678697863463464970, 199592119100636938629838
Offset: 0
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terms=19; A[]=1; Do[A[x]=1/( (1-x)^3 * (1 - x*A[x] - 2*x^2*A'[x]) )+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 17 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=binomial(i+2, 2)+i*sum(j=0, i-1, v[j+1]*v[i-j])); v;
A386266
G.f. A(x) satisfies A(x) = 1/( (1+x) * (1 - x*A(x) - 2*x^2*A'(x)) ).
Original entry on oeis.org
1, 0, 1, 5, 41, 414, 5029, 71154, 1148369, 20813552, 418536501, 9247315593, 222692887441, 5805921628194, 162930550724369, 4897006907712089, 156949282887329569, 5343387198219575688, 192583054168461798805, 7325489802429780722647, 293278021624200874980161
Offset: 0
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terms=21; A[]=1; Do[A[x]=1/( (1+x) * (1 - x*A[x] - 2*x^2*A'[x]) )+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 17 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+i*sum(j=0, i-1, v[j+1]*v[i-j])); v;
Showing 1-5 of 5 results.