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 1-8 of 8 results.

A132685 a(n) = binomial(2^n + 2*n, n).

Original entry on oeis.org

1, 4, 28, 364, 10626, 850668, 218618940, 198773423848, 669741609663270, 8493008777332033900, 405943250253048290447028, 72938914603968404495709630360, 49143490709866058459392200362497820
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2007

Keywords

Links

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), this sequence (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A136555.

Programs

  • Magma
    [Binomial(2^n+2*n,n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    A132695:= n-> binomial(2^n +2*n,n); seq(A132685(n), n=0..20); # G. C. Greubel, Mar 14 2021
  • Mathematica
    Table[Binomial[2^n+2n,n],{n,0,20}] (* Harvey P. Dale, Jun 01 2016 *)
  • PARI
    a(n)=binomial(2^n+2*n,n)
  • Sage
    [binomial(2^n+2*n,n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

a(n) = [x^n] 1/(1-x)^(2^n + n + 1).

A181444 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(2^n*x)^2 )^n/n!.

Original entry on oeis.org

1, 1, 4, 60, 2480, 242296, 53763904, 28363717952, 41396018951936, 215328934357721024, 4740698193856769942528, 430771050114778618253200384, 151994706469390446336698323709952
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 60*x^3 + 2480*x^4 + 242296*x^5 +...
A(x) = 1 + x*[1 + log(A(2x)^2) + log(A(4x)^2)^2/2! + log(A(8x)^2)^3/3! + log(A(16x)^2)^4/4! + log(A(32x)^2)^5/5! +...].
Coefficients in the 2^n-th powers of A(x) begin:
A^(2^0)=[1, 1, 4, 60, 2480, 242296, 53763904, 28363717952,...];
A^(2^1)=[(1), 2, 9, 128, 5096, 490032, 108035840, 56837199680,...];
A^(2^2)=[1,(4), 22, 292, 10785, 1002752, 218139920, 114116667872,...];
A^(2^3)=[1, 8,(60), 760, 24390, 2104632, 444861660, 230028874632,...];
A^(2^4)=[1, 16, 184,(2480), 64540, 4690704, 926901832,...];
A^(2^5)=[1, 32, 624, 10848,(242296), 12359328, 2033807312,...];
A^(2^6)=[1, 64, 2272, 61632, 1568240,(53763904), 5278676128,...];
A^(2^7)=[1, 128, 8640, 414080, 16187360, 588318336,(28363717952),...];
A^(2^8)=[1, 256, 33664, 3040000, 213028800, 12475903232, 658516757120,(41396018951936),...]; ...
where the diagonal terms in parenthesis form this sequence (shift left).
The third column of this table is given by A257272 (observation by _Bruno Berselli_). - _M. F. Hasler_, Apr 27 2015

Crossrefs

Cf. A132695, A181445.

Programs

  • PARI
    {a(n)=local(A=1+sum(m=1,n-1,a(m)*x^m));polcoeff(1+x*sum(m=0,n,log(subst(A^2,x,2^m*x)+x*O(x^n))^m/m!),n)}

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.

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

Views

Author

Paul D. Hanna, Mar 04 2009

Keywords

Examples

			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.

Crossrefs

Cf. A132695, A156905.

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3^(#A-1)))[ #A])); A[n+1]}

Formula

G.f. A(x) satisfies: a(n+1) = [x^n] A(x)^(3^n) for n>=0, with a(0)=1.

A156905 G.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(4^n*x) )^n / n!.

Original entry on oeis.org

1, 1, 4, 184, 69568, 238298048, 10444630574080, 6785507740131518464, 63569094414156159478579200, 8469123482983731772132032235429888, 16127763920864889230376762555181238559440896
Offset: 0

Views

Author

Paul D. Hanna, Mar 04 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 184*x^3 + 69568*x^4 + 238298048*x^5 +...
SERIES REPRESENTATION:
A(x) = 1 + x*[1 + log(A(4x)) + log(A(16x))^2/2! + log(A(64x))^3/3! +...+ log(A(4^n*x))^n/n! +...].
...
GENERATED BY POWERS OF G.F.:
a(n+1) equals the coefficient of x^n in A(x)^(4^n) for n>=0;
the coefficients of A(x)^(4^n) begin:
A^(4^0): [(1), 1, 4, 184, 69568, 238298048, 10444630574080, ...];
A^(4^1): [1, (4), 22, 788, 280625, 954038256, 41781386268864, ...];
A^(4^2): [1, 16, (184), 4464, 1167708, 3830011216, 167171472557448, ...];
A^(4^3): [1, 64, 2272, (69568), 6361840, 15577329728, 669428002912672, ...];
A^(4^4): [1, 256, 33664, 3071744, (238298048), 78858088704, ...];
A^(4^5): [1, 1024, 527872, 182811648, 47958593280, (10444630574080), ...];
In the above table, the diagonal forms this sequence shift left.

Crossrefs

Cf. A132695, A156904.

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(4^(#A-1)))[ #A])); A[n+1]}

Formula

G.f. A(x) satisfies: a(n+1) = [x^n] A(x)^(4^n) for n>=0, with a(0)=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

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.

A181445 G.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(2^n*x))^n/n!.

Original entry on oeis.org

1, 1, 2, 10, 110, 2468, 110678, 9893848, 1765620796, 629689184768, 449004143002080, 640244024341490536, 1825757754872909135396, 10412557570967501281794528, 118766811331255795668894586000
Offset: 0

Views

Author

Paul D. Hanna, Oct 20 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 110*x^4 + 2468*x^5 +...
A(x) = 1 + log(1+x*A(2*x)) + log(1+x*A(4*x))^2/2! + log(1+x*A(8*x))^3/3! + log(1+x*A(16*x))^4/4! + log(1+x*A(32*x))^5/5! +...

Crossrefs

Cf. A132695.

Programs

  • PARI
    {a(n)=local(A=1+sum(m=1,n-1,a(m)*x^m));polcoeff(sum(m=0,n,log(1+x*subst(A,x,2^m*x)+x*O(x^n))^m/m!),n)}
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*subst(A, x, 2^k*x+x*O(x^n))^m)*x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(2^k*x)^n/n!. - Paul D. Hanna, Jan 16 2013

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 1-8 of 8 results.

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