A212029
G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3)^3.
Original entry on oeis.org
1, 1, 3, 21, 190, 2112, 26922, 382110, 5920788, 98862273, 1762572957, 33325846461, 664774457583, 13932829786025, 305788481726799, 7008171327166869, 167321925537782445, 4153009604547937170, 106963758805117459392, 2854029374011293902121, 78773444214057182702790
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 190*x^4 + 2112*x^5 + 26922*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 732*x^4 + 7944*x^5 + 99156*x^6 +..
A(x*A(x)^3) = 1 + x + 6*x^2 + 51*x^3 + 560*x^4 + 7155*x^5 + 102495*x^6 +...
A(x*A(x)^3)^3 = 1 + 3*x + 21*x^2 + 190*x^3 + 2112*x^4 + 26922*x^5 +...
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{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025
A381029
G.f. A(x) satisfies A(x) = 1/(1 - x * A(x*A(x)^2)^2).
Original entry on oeis.org
1, 1, 3, 16, 113, 955, 9178, 97427, 1121705, 13836694, 181295019, 2507119320, 36416096984, 553461581406, 8774534872463, 144744539399484, 2479088917439527, 44004108702467428, 808171916050540308, 15335535608825061803, 300272362335527090277, 6059534345675248667550
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-j+k, j)/(2*n-j+k)*a(n-j, 2*j)));
A381568
G.f. A(x) satisfies A(x) = (1 + x*A(x*A(x)))^2.
Original entry on oeis.org
1, 2, 5, 22, 126, 884, 7149, 64688, 641836, 6888740, 79203860, 968503090, 12525131474, 170555767116, 2436592516874, 36409825487380, 567612675812796, 9211031425896752, 155283809480528788, 2714788300934206360, 49140787009610861896, 919625415852055598804, 17768937720619971300781
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+2*k, j)/(n-j+k)*a(n-j, j)));
A212028
G.f. satisfies: A(x) = 1 + x*A(x*A(x)^3)^2.
Original entry on oeis.org
1, 1, 2, 11, 74, 635, 6296, 70268, 864106, 11546531, 165996792, 2548556963, 41546769324, 715850868468, 12986529841038, 247255748839532, 4926870211273246, 102495266879754087, 2221254395951869988, 50049980203162990978, 1170440788530570387644
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 74*x^4 + 635*x^5 + 6296*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 26*x^3 + 174*x^4 + 1462*x^5 + 14279*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 46*x^3 + 306*x^4 + 2526*x^5 + 24311*x^6 +...
A(x*A(x)^3) = 1 + x + 5*x^2 + 32*x^3 + 273*x^4 + 2715*x^5 + 30542*x^6 + 379200*x^7 + 5117211*x^8 + 74266646*x^9 + 1150267802*x^10 +...
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{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
A212030
G.f. satisfies: A(x) = 1 + x*A(x*A(x)^2)^3.
Original entry on oeis.org
1, 1, 3, 18, 142, 1350, 14607, 174626, 2263749, 31426878, 463144150, 7199095692, 117452998632, 2003613768328, 35628141598164, 658723330672311, 12636278430184303, 251042922016657782, 5156985005918404047, 109382326645948764003, 2392477607054828471286
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 142*x^4 + 1350*x^5 + 14607*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 42*x^3 + 329*x^4 + 3092*x^5 + 33090*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 73*x^3 + 570*x^4 + 5307*x^5 + 56226*x^6 +...
A(x*A(x)^2) = 1 + x + 5*x^2 + 37*x^3 + 346*x^4 + 3745*x^5 + 45132*x^6 +...
A(x*A(x)^2)^3 = 1 + 3*x + 18*x^2 + 142*x^3 + 1350*x^4 + 14607*x^5 +...
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{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^2)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
Showing 1-5 of 5 results.