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 41-50 of 76 results. Next

A339017 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6)).

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 2, 2, 142, 506, 1346, 3170, 53198, 375234, 1880738, 7919082, 72104190, 678488362, 5164781154, 33220643026, 271431061614, 2710340281426, 26278673924322, 228727591600826, 2081516848032222, 21560234032116154, 236863265302626722, 2521687569105476002
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 19 2020

Keywords

Links

Crossrefs

Cf. A001861, A057837, A194689, A339014, A339027.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 27}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1 - x - x^2/2 - x^3/6)))) \\ Michel Marcus, Nov 19 2020

Formula

a(0) = 1; a(n) = 2 * Sum_{k=4..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A057837(k) * A057837(n-k).

A339027 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6 - x^4 / 24)).

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 506, 1850, 5018, 12014, 26886, 1066782, 8193070, 42723722, 185108514, 719359762, 10426744914, 118490840686, 976376930502, 6583701431086, 38977418758494, 377188932759354, 4671829781287922, 51479602726372402, 483303800325691922
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 20 2020

Keywords

Links

Crossrefs

Cf. A001861, A057814, A194689, A339014, A339017.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2 - x^3/6 - x^4/24)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 5, n}]; Table[a[n], {n, 0, 28}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2 * (exp(x) - 1 - x - x^2/2 - x^3/6 - x^4/24)))) \\ Michel Marcus, Nov 20 2020

Formula

a(0) = 1; a(n) = 2 * Sum_{k=5..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A057814(k) * A057814(n-k).

A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

Original entry on oeis.org

1, 2, 4, 10, 32, 114, 448, 1978, 9472, 48738, 270336, 1595114, 9965568, 65852882, 457326592, 3329243546, 25356271616, 201326396098, 1663597019136, 14279558011850, 127044810702848, 1170023757062450, 11136610150121472, 109395885009537402, 1107781178494025728, 11549900930966957346
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2022

Keywords

Links

Crossrefs

Cf. A000557, A000807, A001861.
Cf. A003724, A136630, A352280.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[2 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k * Binomial[n, k] * BellB[k, -1] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 27 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

Formula

E.g.f.: exp( 2 * sinh(x) ).
a(n) = Sum_{k=0..n} 2^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

Original entry on oeis.org

1, 4, 18, 90, 494, 2946, 18926, 130066, 950654, 7353794, 59954638, 513333618, 4601380766, 43062556322, 419742815726, 4252083713874, 44680229906622, 486145710591874, 5468499473222670, 63503107472489266, 760281866742088670, 9373065303624742498, 118858898763010225198
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 25 2022

Keywords

Links

Crossrefs

Cf. A000110, A001861, A035009, A194689, A217924, A293024, A339014.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+2*x], {x, 0, nmax}], x] * Range[0, nmax]!
Table[(BellB[n+2, 2] - BellB[n+1, 2])/4, {n, 0, 25}] (* Vaclav Kotesovec, Jul 21 2025 *)

Formula

a(n) ~ n^(n+2) * exp(n/LambertW(n/2) - n - 2) / (4 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+2)).
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k+1) * Bell(n-k+1). - Ilya Gutkovskiy, Jun 26 2022

A355253 Expansion of e.g.f. exp(2*(exp(x) - 1) - 3*x).

Original entry on oeis.org

1, -1, 3, -5, 19, -29, 171, -69, 2339, 5139, 57563, 303403, 2397011, 17237507, 139011211, 1151110299, 10076637827, 91903924979, 874688607035, 8656097294091, 88932728790195, 946748093175523, 10426787247224043, 118620906668843131, 1392128306377939427, 16833088095308098003
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 26 2022

Keywords

Comments

Inverse binomial transform of A194689.

Links

Crossrefs

Cf. A001861, A217923, A194689, A355252.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Exp[2*Exp[x]-2-3*x], {x, 0, nmax}], x] * Range[0, nmax]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) - 3*x))) \\ Michel Marcus, Dec 04 2023

Formula

a(n) ~ 8 * n^(n-3) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-3)).
a(0) = 1; a(n) = -3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

A036073 Triangle of coefficients arising in calculation of A002872 and A002874 (sorting numbers).

Original entry on oeis.org

1, 2, 1, 5, 1, 6, 15, 1, 11, 30, 52, 1, 20, 80, 150, 203, 1, 37, 210, 525, 780, 877, 1, 70, 560, 1785, 3395, 4263, 4140, 1, 135, 1526, 6125, 14140, 22288, 24556, 21147, 1, 264, 4240, 21420, 58842, 109998, 150402, 149040, 115975, 1, 521, 11970, 76385, 248115
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For connection to A002872, A002874, and other columns of A162663, see the formula in A162663. - Andrey Zabolotskiy, Oct 25 2017

Examples

			Triangle begins:
  1;
  .  2;
  .  1,  5;
  .  1,  6,  15;
  .  1, 11,  30,  52;
  .  1, 20,  80, 150, 203;
  .  1, 37, 210, 525, 780, 877;
  ...

References

  • T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

Links

Crossrefs

Row sums give A001861.
Diagonal gives A000110(n+1) - Alois P. Heinz, Mar 27 2013
Cf. A162663.

Programs

  • Maple
    egf:= exp(exp(x*y)+y*(exp(x)-1)-1):
T:= (n, k)-> n!*coeff(series(coeff(series(egf, y, k+1)
                , y, k), x, n+1), x, n):
seq(seq(T(n, k), k=min(n, 1)..n), n=0..10);  # Alois P. Heinz, Mar 28 2013
  • PARI
    T(n, k) = { my(y = 'y + 'y*O('y^k), x = 'x + 'x*O('x^n); ); n!*polcoeff(polcoeff(exp(exp(x*y)+y*(exp(x)-1)-1), n, 'x), k, 'y); }
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* print triangle */
\\ Michel Marcus, Mar 27 2013
  • PARI
    listpols(n)= {my(z = t + t*O(t^n)); zp = exp(exp(z)-1+(exp(p*z)-1)/p); for (i=0, n, print(i!*polcoeff(zp, i, t)););} \\ Michel Marcus, Mar 27 2013

Formula

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

Extensions

Edited by Vladeta Jovovic, Sep 17 2003
Name corrected by Andrey Zabolotskiy, Oct 22 2017

A068199 One of a family of sequences that interpolates between the Bell numbers and the factorials.

Original entry on oeis.org

1, 2, 6, 24, 114, 618, 3732, 24702, 177126, 1363740, 11195286, 97437138, 894857712, 8637708858, 87333790686, 922203924216, 10144109299146, 115972625504994, 1375221840671220, 16884112119546534, 214270296662325534, 2806600053170775372, 37892025089041181982
Offset: 0

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

References

  • G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

Crossrefs

Cf. A000110, A001861, this, A068200, A068201, ..., A000142.
Equals 2 * A027710(n).

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*g(n-k), k=1..n-1))*3)
    end:
a:= n-> `if`(n=0, 1, 2*g(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 09 2008
  • Mathematica
    a[n_] := 2*BellB[n-1, 3]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 28 2014 *)

Formula

E.g.f.: 1 + 2*exp(3exp(x)-3).

Extensions

More terms from Alois P. Heinz, Oct 09 2008

A068200 One of a family of sequences that interpolates between the Bell numbers and the factorials.

Original entry on oeis.org

1, 2, 6, 24, 120, 696, 4536, 32568, 254136, 2133816, 19130040, 182000952, 1828296888, 19311334200, 213709376184, 2470302259512, 29746381049016, 372270346391352, 4831940144914104, 64925998174811448, 901626111996723384, 12920858504042924856
Offset: 0

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

References

  • G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

Crossrefs

Cf. A000110, A001861, A068199, this, A068201, ..., A000142.

Programs

  • Maple
    g:= proc(n) option remember; `if` (n=0, 1, (1+add(binomial (n-1, k-1) * g(n-k), k=1..n-1)) * 4) end:
a:= n-> `if`(n<=1, n+1, 6*g(n-2)): seq (a(n), n=0..25); # Sean A. Irvine, Feb 02 2024

Extensions

More terms from Sean A. Irvine, Feb 02 2024

A068201 One of a family of sequences that interpolates between the Bell numbers and the factorials.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 4920, 37320, 309120, 2763720, 26440920, 268864320, 2889978120, 32689371720, 387638491920, 4803456571320, 62026296732120, 832595493624720, 11592967532632920, 167119530011233320, 2489936579950221120, 38283169922493447720
Offset: 0

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

References

  • G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

Crossrefs

Cf. A000110, A001861, A068199, A068200, this, ..., A000142.

Programs

  • Maple
    g:= proc(n) option remember; `if` (n=0, 1, (1+add(binomial (n-1, k-1) * g(n-k), k=1..n-1)) * 5) end:
a:= n-> `if`(n<=2, (n+1)!, 24*g(n-3)): seq (a(n), n=0..25); # Sean A. Irvine, Feb 02 2024

Extensions

More terms from Sean A. Irvine, Feb 02 2024

A276506 E.g.f.: exp(9*(exp(x)-1)).

Original entry on oeis.org

1, 9, 90, 981, 11511, 144108, 1911771, 26730981, 392209380, 6016681467, 96202473183, 1599000785730, 27563715220509, 491777630207037, 9064781481234546, 172346601006842337, 3375007346801025099, 67983454804021156548, 1406921223577401454239, 29881379179971835132761
Offset: 0

Views

Author

Vincenzo Librandi, Sep 17 2016

Keywords

Comments

Number of ways of placing n labeled balls into n unlabeled (but 9-colored) boxes.

Links

Crossrefs

Cf. similar sequences with e.g.f. exp(k*(exp(x)-1)): A001861 (k=2), A027710 (k=3), A078944 (k=4), A144180 (k=5) A144223 (k=6), A144263 (k=7), A221159 (k=8), this sequence (k=9), A276507 (k=10).

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*9)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 25 2017
  • Mathematica
    Table[BellB[n, 9], {n, 0, 30}]
  • PARI
    my(x='x+O('x^99)); Vec(serlaplace(exp(9*(exp(x)-1)))) \\ Altug Alkan, Sep 17 2016

Formula

G.f.: A(x) satisfies 9*(x/(1-x))*A(x/(1-x)) = A(x)-1; nine times the binomial transform equals this sequence shifted one place left.
Previous Showing 41-50 of 76 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.