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 1-4 of 4 results.

A078940 Row sums of A078938.

Original entry on oeis.org

1, 4, 19, 103, 622, 4117, 29521, 227290, 1865881, 16239523, 149142952, 1439618143, 14555631781, 153700654036, 1690684883191, 19328770917499, 229203640111870, 2814018686591089, 35711716110387589, 467766675528462562
Offset: 0

Views

Author

Paul D. Hanna, Dec 18 2002

Keywords

Comments

Divide by 3^n and insert an initial 1 to get sequence that shifts left one place under 1/3 order binomial transformation. - Franklin T. Adams-Watters, Jul 13 2006
Binomial transform of A027710. - Vaclav Kotesovec, Jun 26 2022

Links

Crossrefs

Column k=3 of A335975.
Cf. A027710, A035009, A078938, A078945, A355254.

Programs

  • Maple
    A078940 := proc(n) local a,b,i;
a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=3,%),66)) end:
seq(A078940(n),n=0..19); # Peter Luschny, Mar 30 2011
  • Mathematica
    Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
Table[1/E^3/3*Sum[m^n/m!*3^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 3]/3, {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2016 *)
nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - (k+4)*x - 3*(k+1)*x^2/g[k+1]; CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 15 2016, after Sergei N. Gladkovskii *)

Formula

E.g.f.: exp(3*(exp(x)-1)+x).
Stirling transform of [1, 3, 3^2, 3^3, ...]. - Gerald McGarvey, Jun 01 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-3}*f_n(3). - Milan Janjic, May 30 2008
G.f.: 1/T(0), where T(k) = 1 - (k+4)*x - 3*(k+1)*x^2/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2016
a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - Ilya Gutkovskiy, Apr 20 2020
a(n) ~ n^(n+1) * exp(n/LambertW(n/3) - n - 3) / (3 * sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 05 2023

Extensions

More terms from Robert G. Wilson v, Dec 19 2002

A367888 Expansion of e.g.f. exp(3*(exp(x) - 1) - 2*x).

Original entry on oeis.org

1, 1, 4, 13, 61, 304, 1747, 10945, 74830, 550687, 4335109, 36272086, 320980645, 2991373597, 29253607780, 299258487553, 3193634980753, 35469069928792, 409082335024591, 4890313138089133, 60489400453642822, 772967507343358171, 10189818916331129017, 138398721137005215526
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A027710, A126617, A194689, A355254, A367889.

Programs

  • Maple
    b:= proc(n, k, m) option remember; `if`(n=0, 3^m, `if`(k>0,
      b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 29 2025
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-2)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - 2*x))) \\ Michel Marcus, Dec 04 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A027710(k).

A367890 Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 3, 3, 30, 93, 633, 3342, 22809, 156063, 1183872, 9453711, 80455125, 721576560, 6809391111, 67332650007, 695777512638, 7493572404345, 83926492573341, 975467527353750, 11744536832206149, 146234590864310019, 1880198749437144456, 24928860500681953683
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A027710, A194689, A346738, A355253, A355254, A367888, A367891.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 3 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-3)^n / k!.
a(0) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A027710(k).

A367919 Expansion of e.g.f. exp(4*(exp(x) - 1) - x).

Original entry on oeis.org

1, 3, 13, 67, 397, 2627, 19085, 150339, 1272205, 11481155, 109852813, 1109011779, 11765211021, 130707706435, 1516160466573, 18314760232771, 229865470694797, 2991427959247939, 40292570823959693, 560791503840522563, 8053114165521427341, 119158887402348541507
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A078944, A217924, A355254, A367891, A367920, A367921.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[4 (Exp[x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

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

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