cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A257272 a(n) = 2^(n-1)*(2^n+7).

Original entry on oeis.org

4, 9, 22, 60, 184, 624, 2272, 8640, 33664, 132864, 527872, 2104320, 8402944, 33583104, 134275072, 536985600, 2147713024, 8590393344, 34360655872, 137440788480, 549759483904, 2199030595584, 8796107702272, 35184401448960, 140737547075584, 562950070861824, 2251800048566272, 9007199724503040
Offset: 0

Views

Author

M. F. Hasler, Apr 27 2015

Keywords

Comments

For n in A057195, a(n) is of deficiency 8, i.e., in A125247.
Also, the third column (k=2) of the table given in A181444.

Links

Crossrefs

Cf. A000079, A007582, A028403, A256873, A257273, A256871.
Cf. A168415, A057195, A104066, A125247.

Programs

  • Magma
    [2^(n-1)*(2^n+7): n in [0..25]]; // Vincenzo Librandi, Apr 27 2015
  • Mathematica
    Table[2^(n - 1) (2^n + 7), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(4 - 15 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)
  • PARI
    a(n)=2^(n-1)*(2^n+7)
  • PARI
    Vec((4-15*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

Formula

a(n) = 2^(n-1)*A168415(n).
n in A057195 <=> A168415(n) in A104066 <=> a(n) in A125247.
G.f.: (4-15*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

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

Views

Author

Paul D. Hanna, Oct 16 2023

Keywords

Comments

In general, we have the following identity:
given A(x) = Sum_{n>=0} a(n)*x^n satisfies
A(x) = 1 + x*Sum_{n>=0} p^n * log( A(q^n*x) )^n / n!,
then a(n+1) = [x^n] A(x)^(p*q^n) for n >= 0, with a(0)=1,
for arbitrary fixed parameters p and q.
Here, p = 2 and q = 3.

Examples

			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.

Links

Crossrefs

Cf. A132695, A156904, A156905, A181444.

Programs

  • PARI
    {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),", "))
  • PARI
    {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),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*Sum_{n>=0} 2^n*log( A(3^n*x) )^n / n!.
(2) a(n+1) = [x^n] A(x)^(2*3^n) for n >= 0, with a(0)=1.

A181588 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(2^n*x) )^n/(2^n*n!).

Original entry on oeis.org

1, 1, 1, 3, 28, 658, 36336, 4918960, 1913308480, 2722820397008, 16147361055588096, 383879833447290828032, 34790330152647391683716096, 11818617432918372433417869737472
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 28*x^4 + 658*x^5 + 36336*x^6 +...
A(x) = 1 + x*[1 + log(A(2x))/2 + log(A(4x))^2/(4*2!) + log(A(8x))^3/(8*3!) + log(A(16x))^4/(16*4!) +...].
Coefficients in the 2^n-th powers of A(x) begin:
A^(2^0)=[1,(1), 1, 3, 28, 658, 36336, 4918960, 1913308480,...];
A^(2^1)=[1, 2,(3), 8, 63, 1378, 74053, 9912076, 3836532284,...];
A^(2^2)=[1, 4, 10,(28), 167, 3056, 154060, 20129640, 7713183207,...];
A^(2^3)=[1, 8, 36, 136,(658), 8008, 336692, 41562232, 15590683759,...];
A^(2^4)=[1, 16, 136, 848, 4788,(36336), 867384, 89267088,...];
A^(2^5)=[1, 32, 528, 6048, 55208, 456544,(4918960), 224294304,...];
A^(2^6)=[1, 64, 2080, 45888, 776272, 10833088, 133934688,(1913308480), ...]; ...
where the diagonal terms in parenthesis form this sequence (shift left).

Crossrefs

Cf. A132695, A181444.

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(2^(#A-2)))[ #A])); A[n+1]}

Formula

a(n+1) = [x^n] A(x)^(2^(n-1)) for n>=0, with a(0)=1, where A(x) = Sum_{n>=0} a(n)*x^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

Views

Author

Paul D. Hanna, Oct 17 2023

Keywords

Comments

In general, we have the following identity:
given A(x) = Sum_{n>=0} a(n)*x^n satisfies
A(x) = 1 + x*Sum_{n>=0} p^n * log( A(q^n*x) )^n / n!,
then a(n+1) = [x^n] A(x)^(p*q^n) for n >= 0, with a(0)=1,
for arbitrary fixed parameters p and q.
Here, p = 3 and q = 2.

Examples

			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.

Crossrefs

Cf. A132695, A156904, A156905, A181444, A366226.

Programs

  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.
(2) a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0, with a(0)=1.
Showing 1-4 of 4 results.

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