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.

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A367939 Expansion of e.g.f. exp(exp(4*x) - 1 - 2*x).

Original entry on oeis.org

1, 2, 20, 168, 1936, 25376, 378688, 6284928, 114471168, 2263605760, 48192279552, 1097180784640, 26562251100160, 680591327567872, 18381995707154432, 521521320660205568, 15495495061984051200, 480873815489757970432, 15549555768325162926080, 522810678067316117733376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A124311, A126617, A355162, A367786, A367938, A367940.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-2)^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - 4*x)) / (1 - 4*x) ).
a(n) = exp(-1) * Sum_{k>=0} (4*k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n+k) * Bell(k).
a(n) = 2^n * |A124311(n)|.

A367940 Expansion of e.g.f. exp(exp(4*x) - 1 - 3*x).

Original entry on oeis.org

1, 1, 17, 113, 1377, 17185, 252401, 4104721, 73500609, 1430779713, 30026750161, 674586467505, 16130795165473, 408560492670049, 10915540174130353, 306531211899158609, 9019774516570506113, 277345675943850865281, 8889954225208868308369, 296408283056785166556401
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A124311, A355163, A346738, A367786, A367939.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-3)^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 3*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( 3 * A(x) - 4 * A(x/(1 - 4*x)) / (1 - 4*x) ).
a(n) = exp(-1) * Sum_{k>=0} (4*k-3)^n / k!.
a(0) = 1; a(n) = -3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * 4^k * Bell(k).

A367945 Expansion of e.g.f. exp(2*(exp(2*x) - 1) - x).

Original entry on oeis.org

1, 3, 17, 115, 929, 8547, 87729, 988883, 12100929, 159331523, 2241395537, 33493315379, 529089873121, 8799587162659, 153545747910129, 2802447872764307, 53358770299683457, 1057354788073681283, 21760656533457251985, 464240718007022020083, 10249389749356980403745
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000296, A124311, A217924, A308543, A367946.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[2 (Exp[2 x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(2*x) - 1) - x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( A(x) - 4 * A(x/(1 - 2*x)) / (1 - 2*x) ).
a(n) = exp(-2) * Sum_{k>=0} 2^k * (2*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^(k+1) * a(n-k).
Previous Showing 11-13 of 13 results.

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