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 1-7 of 7 results.

A328182 Expansion of e.g.f. 1 / (2 - exp(3*x)).

Original entry on oeis.org

1, 3, 27, 351, 6075, 131463, 3413907, 103429791, 3581223435, 139498558263, 6037616347587, 287444492409231, 14929010774254395, 839982382565841063, 50897213545996785267, 3304312091004451756671, 228821504027595115886955, 16836102104577636004291863, 1311625494765417347634022947
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 06 2019

Keywords

Crossrefs

Cf. A000670, A216794, A247452, A328183.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*3^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2019
  • Mathematica
    nmax = 18; CoefficientList[Series[1/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[3^k Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[3^n HurwitzLerchPhi[1/2, -n, 0]/2, {n, 0, 18}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} 3^k * binomial(n,k) * a(n-k).
a(n) = Sum_{k>=0} (3*k)^n / 2^(k + 1).
a(n) = 3^n * A000670(n).
a(n) ~ n! * 3^n / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A367785 Expansion of e.g.f. exp(exp(3*x) - x - 1).

Original entry on oeis.org

1, 2, 13, 89, 772, 7745, 87949, 1109288, 15332539, 229840361, 3706130914, 63857565095, 1169261937973, 22646779177898, 462143532144937, 9902312863237637, 222119823632283628, 5202170552214520637, 126914730275907871201, 3218552632981994910248, 84686139239808135094879, 2307953474037054591248501
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2023

Keywords

Crossrefs

Cf. A000110, A000296, A124311, A247452, A284859, A284860, A330605, A367744, A367786.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - x - 1))) \\ Michel Marcus, Nov 30 2023

Formula

a(n) = exp(-1) * Sum_{k>=0} (3*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * Bell(k).

A336636 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^3 - 1).

Original entry on oeis.org

1, 3, 33, 660, 20817, 935388, 56149098, 4311694467, 410200118577, 47174279349540, 6431874002292978, 1023398757621960327, 187566773426941146498, 39164789611542644630415, 9229712819952662426436507, 2435069724188535096598261305
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

Crossrefs

Cf. A002893, A023998, A247452, A336635, A336637.

Programs

  • Mathematica
    nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^3 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * A002893(k) * k * a(n-k).

A351892 Expansion of e.g.f. exp( sinh(sqrt(3)*x) / sqrt(3) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 40, 205, 952, 4921, 31168, 189145, 1318528, 9843781, 74869888, 632536933, 5475991552, 49996774897, 485393809408, 4829958877105, 50858117779456, 554544498995965, 6259096187060224, 73822470722135293, 894846287081242624, 11261265009125680681, 146272258394568687616
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

Crossrefs

Cf. A003724, A009229, A136630, A247452, A351891, A381277.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[3] x]/Sqrt[3]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 3^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 3^k * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025

A292913 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(k*x)-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 15, 0, 1, 5, 32, 135, 240, 52, 0, 1, 6, 50, 320, 1215, 1664, 203, 0, 1, 7, 72, 625, 3840, 12636, 12992, 877, 0, 1, 8, 98, 1080, 9375, 53248, 147987, 112256, 4140, 0, 1, 9, 128, 1715, 19440, 162500, 831488, 1917999, 1059840, 21147, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2017

Keywords

Examples

			E.g.f. of column k: A_k(x) =  1 + k*x/1! + 2*k^2*x^2/2! + 5*k^3*x^3/3! + 15*k^4 x^4/4! + 52*k^5*x^5/5! + ...
Square array begins:
1,   1,     1,      1,      1,       1,  ...
0,   1,     2,      3,      4,       5,  ...
0,   2,     8,     18,     32,      50,  ...
0,   5,    40,    135,    320,     625,  ...
0,  15,   240,   1215,   3840,    9375,  ...
0,  52,  1664,  12636,  53248,  162500,  ...

Crossrefs

Columns k=0..3 give A000007, A000110, A055882, A247452.
Rows n=0..2 give A000012, A001477, A001105.
Main diagonal gives A292914.

Programs

  • Maple
    A:= (n, k)-> k^n * combinat[bell](n):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 26 2017
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[Exp[k x] - 1], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-((-1)^(i + 1) (i - 1) + i + 3) k x/4, 1, {i, 0, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

O.g.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - k*x/(1 - 3*k*x/(1 - k*x/(1 - 4*k*x/(1 - ...))))))))), a continued fraction.
E.g.f. of column k: exp(exp(k*x)-1).
A(n,k) = exp(-1)*k^n*Sum_{j>=0} j^n/j!.
A(n,k) = k^n * Bell(n). - Alois P. Heinz, Sep 26 2017

A367938 Expansion of e.g.f. exp(exp(3*x) - 1 - 2*x).

Original entry on oeis.org

1, 1, 10, 55, 487, 4654, 51463, 632125, 8536492, 125279785, 1981246555, 33530245984, 603797462677, 11513675558701, 231539488842610, 4893151984630579, 108334206855000739, 2505977899186557502, 60419653270442268643, 1515077412621445514089, 39437350309301393464876, 1063746973172416765272589
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A124311, A126617, A247452, A284864, A367785, A367888.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[3 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] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] (-2)^(n - k) 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - 3*x)) / (1 - 3*x) ).
a(n) = exp(-1) * Sum_{k>=0} (3*k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * 3^k * Bell(k).

A369783 Expansion of e.g.f. exp( exp(3*(exp(x)-1))-1 ).

Original entry on oeis.org

1, 3, 21, 192, 2154, 28434, 429213, 7261788, 135698268, 2769463335, 61186736415, 1452889463034, 36857766745749, 993941679586098, 28370018078000985, 853903169641805925, 27014392815958815969, 895723118730738795837, 31048284069527339602902
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Crossrefs

Cf. A000258, A346417.
Cf. A000110, A247452.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(exp(3*(exp(x)-1))-1)))

Formula

a(n) = Sum_{k=0..n} 3^k * Stirling2(n,k) * Bell(k) = Sum_{k=0..n} Stirling2(n,k) * A247452(k).
Showing 1-7 of 7 results.

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