A156904
G.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(3^n*x) )^n / n!.
Original entry on oeis.org
1, 1, 3, 63, 6732, 3414312, 10221878106, 243813944182248, 50538758405328815616, 87376772859536771916909012, 1235009698863206337006094872463887, 142641072494398006081741872595533545306244
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 63*x^3 + 6732*x^4 + 3414312*x^5 +...
SERIES REPRESENTATION:
A(x) = 1 + x*[1 + log(A(3x)) + log(A(9x))^2/2! + log(A(27x))^3/3! +...+ log(A(3^n*x))^n/n! +...].
...
GENERATED BY POWERS OF G.F.:
a(n+1) equals the coefficient of x^n in A(x)^(3^n) for n>=0;
the coefficients of A(x)^(3^n) begin:
A^(3^0): [(1), 1, 3, 63, 6732, 3414312, 10221878106, ...];
A^(3^1): [1, (3), 12, 208, 20610, 10284678, 30686274630, ...];
A^(3^2): [1, 9, (63), 867, 66330, 31246902, 92246164932, ...];
A^(3^3): [1, 27, 432, (6732), 273024, 97968096, 278472473082, ...];
A^(3^4): [1, 81, 3483, 109863, (3414312), 385422948, 853280745822, ...];
A^(3^5): [1, 243, 30132, 2553768, 168586110, (10221878106), ...];
In the above table, the diagonal forms this sequence shift left.
-
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3^(#A-1)))[ #A])); A[n+1]}
A366226
O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 2^n * log( A(3^n*x) )^n / n!.
Original entry on oeis.org
1, 1, 6, 261, 56070, 56526498, 334429044030, 15777272891508021, 6500948711591606135796, 22416650201723925643982814186, 632905244163070372226486183732882316, 146120187946706698644410320973489902454862324, 277121097159744219425840626808464318501357604841881466
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 261*x^3 + 56070*x^4 + 56526498*x^5 + 334429044030*x^6 + 15777272891508021*x^7 + 6500948711591606135796*x^8 + ...
where
A(x) = 1 + x*[1 + 2*log(A(3*x)) + 2^2*log(A(3^2*x))^2/2! + 2^3*log(A(3^2*x))^3/3! + ... + 2^n*log(A(3^n*x))^n/n! + ...].
RELATED SERIES.
log(A(x)) = x + 11*x^2/2 + 766*x^3/3 + 223187*x^4/4 + 282345766*x^5/5 + 2006233236098*x^6/6 + 110438567161208518*x^7/7 + ...
RELATED TABLE.
The table of coefficients of x^k in A(x)^(2*3^n) begins:
n=0: [1, 2, 13, 534, 112698, 113168268, ...];
n=1: [1, 6, 51, 1766, 345165, 340906254, ...];
n=2: [1, 18, 261, 7350, 1112382, 1035922644, ...];
n=3: [1, 54, 1755, 56070, 4589001, 3250238022, ...];
n=4: [1, 162, 14013, 894294, 56526498, 12817431900, ...];
n=5: [1, 486, 120771, 20555046, 2731197285, 334429044030, ...]; ...
in which the main diagonal equals this sequence shift left,
illustrating that a(n+1) = [x^n] A(x)^(2*3^n) for n >= 0.
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{a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(2*3^(#A-1)))[ #A])); A[n+1]}
for(n=0,15,print1(a(n),", "))
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{a(n) = my(A=1+x); for(i=1, n, A = 1 + x*sum(m=0,#A, 2^m*log( subst(Ser(A),x,3^m*x +x*O(x^n)))^m/m!) ); polcoeff(A,n)}
for(n=0,15,print1(a(n),", "))
A366227
O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.
Original entry on oeis.org
1, 1, 6, 138, 8648, 1272948, 424058592, 334836466656, 728593565874816, 5632989888855720864, 184539760855097635059200, 25027477244647424010315231744, 13206715998089387470949589465286656, 26431031766456352400292737393044784872448, 199091399877503863934385670788355318673030504448
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 138*x^3 + 8648*x^4 + 1272948*x^5 + 424058592*x^6 + 334836466656*x^7 + 728593565874816*x^8 + ...
where
A(x) = 1 + x*[1 + 3*log(A(2*x)) + 3^2*log(A(2^2*x))^2/2! + 3^3*log(A(2^2*x))^3/3! + ... + 3^n*log(A(2^n*x))^n/n! + ...].
RELATED SERIES.
log(A(x)) = x + 11*x^2/2 + 397*x^3/3 + 33991*x^4/4 + 6318201*x^5/5 + 2536406543*x^6/6 + 2340834765809*x^7/7 + ...
RELATED TABLE.
The table of coefficients of x^k in A(x)^(3*2^n) begins:
n=0: [1, 3, 21, 451, 26898, 3876222, ...];
n=1: [1, 6, 51, 1028, 56943, 7932774, ...];
n=2: [1, 12, 138, 2668, 128823, 16653720, ...];
n=3: [1, 24, 420, 8648, 340722, 37135560, ...];
n=4: [1, 48, 1416, 37456, 1272948, 97890096, ...];
n=5: [1, 96, 5136, 210848, 8146728, 424058592, ...];
n=6: [1, 192, 19488, 1407808, 83154768, 4578119616, 334836466656, ...]; ...
in which the main diagonal equals this sequence shift left,
illustrating that a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0.
-
{a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3*2^(#A-1)))[ #A])); A[n+1]}
for(n=0, 15, print1(a(n), ", "))
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{a(n) = my(A=1+x); for(i=1, n, A = 1 + x*sum(m=0, #A, 3^m*log( subst(Ser(A), x, 2^m*x +x*O(x^n)))^m/m!) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
Showing 1-3 of 3 results.
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