A386211
G.f. A(x) satisfies A(x) = 1/(1-x)^2 + x^2*A(x)*A'(x).
Original entry on oeis.org
1, 2, 5, 18, 89, 556, 4127, 35084, 334049, 3510574, 40300769, 501455462, 6721438253, 96561557816, 1480441163151, 24132225315816, 416852189961737, 7607668036964506, 146296367990498941, 2957053490913146762, 62682940163232269033, 1390605993609167492932
Offset: 0
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terms = 22; A[] = 1; Do[A[x] = 1/(1-x)^2+x^2*A[x]A'[x] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=i+1+(i-1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;
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;
A386212
G.f. A(x) satisfies A(x) = 1/(1-x)^3 + x^2*A(x)*A'(x).
Original entry on oeis.org
1, 3, 9, 37, 207, 1455, 12073, 113949, 1196499, 13778155, 172269777, 2321464773, 33524717911, 516428180631, 8453096463321, 146532991389613, 2682216423470763, 51706945300407315, 1047284621276095729, 22237895367773398821, 494041637873385734127, 11462206075715032723903
Offset: 0
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=binomial(i+2, 2)+(i-1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;
A385844
G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^3*A''(x))).
Original entry on oeis.org
1, 1, 1, 3, 21, 273, 5737, 177919, 7651849, 436186313, 31842549569, 2897710853939, 321648004495773, 42779331295225353, 6716367934603667145, 1229096733282700520799, 259339594018913458094865, 62500870590534491566841265, 17062742827503910747790541249, 5238263128497776755775631825219
Offset: 0
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terms = 20; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^3*A''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 10 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, sum(k=1, 2, stirling(2, k, 1)*j^k)*v[j+1]*v[i-j])); v;
A385845
G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^4*A'''(x))).
Original entry on oeis.org
1, 1, 1, 1, 7, 175, 10675, 1291675, 272543461, 91847148373, 46382810082589, 33442006088446669, 33141028037446336195, 43779298038683546954491, 75169054733013247990186039, 164244384592052866115015051119, 448551414321306169623754824645385
Offset: 0
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terms = 17; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^4*A'''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 10 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, sum(k=1, 3, stirling(3, k, 1)*j^k)*v[j+1]*v[i-j])); v;
A385846
G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^5*A''''(x))).
Original entry on oeis.org
1, 1, 1, 1, 1, 25, 3025, 1092025, 918393025, 1543818675025, 4670051491951201, 23541729570926148241, 186474039931306081488961, 2215498068423847604734793641, 38020162352221648825602734209201, 913434400512125113270449340963296649, 29925024395177730837015182640209851847809
Offset: 0
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terms = 17; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^5*A''''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 10 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, sum(k=1, 4, stirling(4, k, 1)*j^k)*v[j+1]*v[i-j])); v;
A386239
G.f. A(x) satisfies A(x) = 1/( (1+x) * (1 - x^2*A'(x)) ).
Original entry on oeis.org
1, -1, 0, 0, 1, 3, 11, 47, 253, 1651, 12610, 109744, 1069355, 11520785, 135906642, 1741702304, 24089599321, 357592702647, 5669840845462, 95623921546478, 1709172658222253, 32271612381443479, 641820925099092985, 13410242452064469153, 293676423537521878381
Offset: 0
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terms = 25; A[] = 1; Do[A[x] = 1/((1+x)(1-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+(i-1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;