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).
-
{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),", "))
A352706
G.f. A(x) satisfies: (1 - x*A(x))^7 = 1 - 7*x - x^7*A(x^7).
Original entry on oeis.org
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 + 3*x + 13*x^2 + 65*x^3 + 351*x^4 + 1989*x^5 + 11650*x^6 + 69900*x^7 + 427167*x^8 + 2648438*x^9 + 16612947*x^10 + ...
where
(1 - x*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 + ...
also
(1 - 7*x - x^7*A(x^7))^(1/7) = 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 + ...
which equals 1 - x*A(x).
-
{a(n) = my(A=1+3*x); for(i=1,n,
A = (1 - (1 - 7*x - x^7*subst(A,x,x^7) + x*O(x^(n+1)))^(1/7))/x);
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A376224
G.f. A(x) satisfies A( (x + 3*A(x)^2)^3 ) = A(x)^3.
Original entry on oeis.org
1, 3, 18, 136, 1152, 10458, 99473, 978480, 9872181, 101598389, 1062382809, 11255336235, 120555453344, 1303305334704, 14202627395202, 155847144409224, 1720542786453765, 19096869133735155, 212977164179543266, 2385405242723601582, 26820428322385799784, 302611771988083401990
Offset: 1
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 136*x^4 + 1152*x^5 + 10458*x^6 + 99473*x^7 + 978480*x^8 + 9872181*x^9 + 101598389*x^10 + ...
where A( (x + 3*A(x)^2)^3 ) = A(x)^3.
RELATED SERIES.
A(x)^3 = x^3 + 9*x^4 + 81*x^5 + 759*x^6 + 7362*x^7 + 73386*x^8 + 747567*x^9 + 7749720*x^10 + 81500094*x^11 + 867420469*x^12 + ...
( x^2*A(x) )^(1/3) = x + x^2 + 5*x^3 + 35*x^4 + 284*x^5 + 2508*x^6 + 23401*x^7 + 226950*x^8 + 2265015*x^9 + 23110418*x^10 + ...
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - 3*x^2 - x^4 - x^7 - 2*x^10 - 4*x^13 - 9*x^16 - 22*x^19 - 55*x^22 - 142*x^25 - 376*x^28 - ... + -A352702(n)*x^(3*n+4) + ...
where B(x) = x*(1 - x*G(x))^3 and B(x) = x - 3*x^2 - x^4*G(x^3), where G(x) is the g.f. of A352702 and begins:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 55*x^6 + 142*x^7 + 376*x^8 + 1011*x^9 + 2758*x^10 + ...
-
{a(n) = my(A = x+x^2); for(m=1, n, A = truncate(A) + x^2*O(x^m); A = subst(A, x, (x + 3*A^2)^3 )^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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