A107092
G.f. A(x) satisfies A(x)^3 = A(x^3) + 3*x.
Original entry on oeis.org
1, 1, -1, 2, -4, 9, -22, 55, -142, 376, -1011, 2758, -7614, 21220, -59630, 168759, -480533, 1375676, -3957075, 11430582, -33144264, 96434321, -281447954, 823734157, -2417092933, 7109265120, -20955593252, 61893804180, -183148075432, 542885589115, -1611809502764, 4792612539375
Offset: 0
A(x)^3 = 1 + 3*x + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18 +...
A(x^3) = 1 + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18+...
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{a(n)=local(A=1+x);for(i=1,n,A=(subst(A,x,x^3)+3*x+x*O(x^n))^(1/3)); polcoeff(A,n,x)}
A352704
G.f. A(x) satisfies: (1 - x*A(x))^5 = 1 - 5*x - x^5*A(x^5).
Original entry on oeis.org
1, 2, 6, 21, 80, 320, 1326, 5637, 24434, 107542, 479196, 2157045, 9792702, 44780606, 206055346, 953305632, 4431463863, 20686696920, 96931500840, 455722378776, 2149086843549, 10162544469252, 48176923330632, 228913129263389, 1089973058779915, 5199987220813564
Offset: 0
G.f.: A(x) = 1 + 2*x + 6*x^2 + 21*x^3 + 80*x^4 + 320*x^5 + 1326*x^6 + 5637*x^7 + 24434*x^8 + 107542*x^9 + 479196*x^10 + ...
where
(1 - x*A(x))^5 = 1 - 5*x - x^5 - 2*x^10 - 6*x^15 - 21*x^20 - 80*x^25 - 320*x^30 - 1326*x^35 - 5637*x^40 - 24434*x^45 - 107542*x^50 + ...
also
(1 - 5*x - x^5*A(x^5))^(1/5) = 1 - x - 2*x^2 - 6*x^3 - 21*x^4 - 80*x^5 - 320*x^6 - 1326*x^7 - 5637*x^8 - 24434*x^9 - 107542*x^10 + ...
which equals 1 - x*A(x).
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{a(n) = my(A=1+2*x); for(i=1,n,
A = (1 - (1 - 5*x - x^5*subst(A,x,x^5) + x*O(x^(n+1)))^(1/5))/x);
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A352705
G.f. A(x) satisfies: A(x)^7 = A(x^7) + 7*x.
Original entry on oeis.org
1, 1, -3, 13, -65, 351, -1989, 11650, -69900, 427167, -2648438, 16612947, -105215448, 671760933, -4318468134, 27926126553, -181520036178, 1185220461867, -7769787812787, 51117085998498, -337373170647840, 2233091755252871, -14819626692452231, 98582852467595847
Offset: 0
G.f.: A(x) = 1 + x - 3*x^2 + 13*x^3 - 65*x^4 + 351*x^5 - 1989*x^6 + 11650*x^7 - 69900*x^8 + 427167*x^9 - 2648438*x^10 + ...
such that A(x)^7 = A(x^7) + 7*x, as illustrated by:
A(x)^7 = 1 + 7*x + x^7 - 3*x^14 + 13*x^21 - 65*x^28 + 351*x^35 - 1989*x^42 + 11650*x^49 - 69900*x^56 + 427167*x^63 - 2648438*x^70 + ...
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{a(n) = my(A=1+x); for(i=1,n,
A = (subst(A,x,x^7) + 7*x + x*O(x^n))^(1/7));
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A370545
Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).
Original entry on oeis.org
1, 1, 4, 21, 125, 801, 5388, 37518, 268109, 1955000, 14487754, 108794169, 826054062, 6331064385, 48914088750, 380555960864, 2978892961194, 23444095375593, 185394136871818, 1472396312841250, 11739089289817538, 93921736129064325, 753845680317416682, 6068255413854119432
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 125*x^5 + 801*x^6 + 5388*x^7 + 37518*x^8 + 268109*x^9 + 1955000*x^10 + 14487754*x^11 + 108794169*x^12 + ...
where A(x)^5 = A( x^5 + 5*A(x)^6 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 30*x^7 + 195*x^8 + 1330*x^9 + 9376*x^10 + 67720*x^11 + ...
A(x)^6 = x^6 + 6*x^7 + 39*x^8 + 266*x^9 + 1875*x^10 + 13542*x^11 + 99654*x^12 + ...
Let B(x) be the series reversion of A(x), A(B(x)) = x, which begins
B(x) = x - x^2 - 2*x^3 - 6*x^4 - 21*x^5 - 80*x^6 - 320*x^7 - 1326*x^8 - 5637*x^9 - 24434*x^10 - ... + (-1)^(n-1)*A352703(n-1)*x^n + ...
then B(x)^5 + 5*x^6 = B(x^5).
Let C(x) = x^2/B(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 802*x^7 + 3553*x^8 + 15994*x^9 + 72886*x^10 + ... + A091200(n-1)*x^n + ...
where A(x^2/C(x)) = x and C(A(x)) = A(x)^2/x,
then C(x)^5 = C(x^5)/(1 - 5*C(x^5)/x^4).
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{a(n) = my(A=x+x^2); for(m=1, n, A=truncate(A); A = subst(A, x, x^5 + 5*A^6 +x^5*O(x^m))^(1/5) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
Showing 1-4 of 4 results.
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