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.

Previous Showing 11-18 of 18 results.

A308645 Expansion of e.g.f. exp(1 + x - exp(2*x)).

Original entry on oeis.org

1, -1, -3, 3, 41, 87, -571, -5701, -14575, 156655, 2094925, 9148851, -63364423, -1474212665, -11494853995, 10945362411, 1520718442785, 20719421344991, 100137575499165, -1638818071763869, -45333849658449847, -512404024891840969, -577060092568365467, 99142586163648127771
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 13 2019

Keywords

Crossrefs

Cf. A000587, A055882, A126390, A308536.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[1 + x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Exp[1] Sum[(-1)^k (2 k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 23}]
Table[Sum[Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 23}]

Formula

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(2*k + 1)^n/k!.
a(n) = Sum_{k=0..n} binomial(n,k)*2^k*A000587(k).

A336635 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^2 - 1).

Original entry on oeis.org

1, 2, 14, 176, 3470, 96792, 3590048, 169686792, 9903471502, 696692504552, 57958925154584, 5614276497440712, 625153195794408608, 79159558899671117896, 11293672011942106846808, 1801015209162807119535216, 318805481931592799427378062
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

Crossrefs

Cf. A000984, A023998, A055882, A323666, A336636, A336637.

Programs

  • Mathematica
    nmax = 16; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^2 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[2 k, k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

Formula

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

A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

Original entry on oeis.org

1, 0, 1, 4, 14, 66, 397, 2626, 18797, 148238, 1281134, 11943790, 118998365, 1262189748, 14203022537, 168835162632, 2111832477426, 27708387132906, 380355066174121, 5449577398256414, 81316095965242989, 1261149374033472626, 20293627142875917978, 338263983223664609198
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2021

Keywords

Comments

Exponential transform of A000295.

Crossrefs

Cf. A000295, A000296, A003725, A055882, A143405, A347434, A347435.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000295(k) * a(n-k).
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A003725(k) * A143405(n-k).
a(n) ~ n^(n + 1/2) * (exp(exp(r)*(exp(r) - r - 1) - r/2 - n) / (r^(n + 1/2) * sqrt(2*exp(r)*(1 + 2*r) - (2 + r*(4 + r))))), where r = LambertW(n)/2 + (4 + LambertW(n)) * LambertW(n)^(3/2) / (8 * sqrt(n) * (1 + LambertW(n))). - Vaclav Kotesovec, Jul 07 2022

A347434 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2) ).

Original entry on oeis.org

1, 0, 0, 1, 5, 16, 52, 274, 1990, 14354, 99704, 730225, 6061013, 56151330, 551040830, 5597109717, 59324775741, 664973687438, 7891158217876, 98253448977890, 1273082291906394, 17124091446383666, 239333235895599762, 3476600533730954761, 52394273274018321421
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2021

Keywords

Comments

Exponential transform of A002662.

Crossrefs

Cf. A002662, A006505, A055882, A143405, A347432, A347435.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j*(j+1)/2-1), j=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Sep 02 2021
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x - x^2/2)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - 1 - k (k + 1)/2) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002662(k) * a(n-k).

A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

Original entry on oeis.org

1, 0, 0, 0, 1, 6, 22, 64, 198, 1138, 10004, 83920, 617993, 4226028, 30103686, 251883012, 2490287821, 26456763078, 281404300348, 2966101610920, 31877462564554, 362624252399566, 4437794875670072, 57612897938229380, 773900876490016325, 10599854900351622752
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2021

Keywords

Comments

Exponential transform of A002663.

Crossrefs

Cf. A002663, A055882, A057837, A143405, A347432, A347434.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j^3/6-5*j/6-1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x - x^2/2 - x^3/6)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - 1 - k (k^2 + 5)/6) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002663(k) * a(n-k).

A351891 Expansion of e.g.f. exp( sinh(sqrt(2)*x) / sqrt(2) ).

Original entry on oeis.org

1, 1, 1, 3, 9, 25, 105, 443, 1969, 10609, 57265, 338547, 2190969, 14498185, 104277849, 784965803, 6150938593, 51229928929, 440694547681, 3967606065891, 37247506348905, 361022009762809, 3645855348771273, 38001754007842715, 409302848055407761, 4558828622414199121
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

Crossrefs

Cf. A003724, A009229, A055882, A136630, A351892, A381277.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[2] x]/Sqrt[2]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 2^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) * 2^k * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025

A055883 Exponential transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 5, 15, 15, 5, 15, 60, 90, 60, 15, 52, 260, 520, 520, 260, 52, 203, 1218, 3045, 4060, 3045, 1218, 203, 877, 6139, 18417, 30695, 30695, 18417, 6139, 877, 4140, 33120, 115920, 231840, 289800, 231840, 115920, 33120, 4140, 21147
Offset: 0

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, ...] DELTA [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

Examples

			   1;
   1,  1;
   2,  4,  2;
   5, 15, 15,  5;
  15, 60, 90, 60, 15;
  ...

Links

Crossrefs

Cf. A000110, A007318, A056860.
Row sums give A055882.

Programs

  • Mathematica
    T[ n_, k_] := Binomial[n, k] * BellB[n]; (* Michael Somos, Apr 09 2025 *)
  • PARI
    T(n, k) = binomial(n, k) * sum(j=0, n, stirling(n, j, 2)); /* Michael Somos, Apr 09 2025 */

Formula

a(n,k) = Bell(n)*C(n,k).
E.g.f.: A(x,y) = exp(exp(x+xy)-1).

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
Previous Showing 11-18 of 18 results.

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