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 1-3 of 3 results.

A126617 a(n) = Sum_{i=0..n} (-2)^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n).

Original entry on oeis.org

1, -1, 2, -3, 7, -10, 31, -21, 204, 307, 2811, 12100, 74053, 432211, 2768858, 18473441, 129941283, 956187814, 7351696139, 58897405759, 490681196604, 4242903803727, 38014084430983, 352341755256348, 3373662303816313, 33326335433122711, 339232538387804530
Offset: 0

Views

Author

N. J. A. Sloane, Aug 04 2007

Keywords

Comments

a(n) is positive starting at n=8. - Karol A. Penson and Olivier Gérard, Oct 22 2007
Hankel transform is A000178. - Paul Barry, Apr 23 2009

Examples

			G.f.: 1 - 1*x + 2*x^2 - 3*x^3 + 7*x^4 - 10*x^5 + 31*x^6 - 21*x^7 + 204*x^8 + 307*x^9 + 2811*x^10 + 12100*x^11 + 74053*x^12 + 432211*x^13 + ...

Links

Crossrefs

Cf. A000110, A000296, A005493, A124311, A126390, A153732.

Programs

  • Mathematica
    Table[ Sum[ (-2)^(n - k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *)
With[{nn=30},CoefficientList[Series[Exp[Exp[x]-2x-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 19 2025 *)

Formula

E.g.f.: exp(exp(x)-2*x-1). - Vladeta Jovovic, Aug 04 2007
a(n) = e^(-1) * Sum_{k>=0} (k-2)^n / k!. This is a Dobinski-type formula. - Karol A. Penson and Olivier Gérard, Oct 22 2007
G.f.: 1/(1+x-x^2/(1-2x^2/(1-x-3x^2/(1-2x-4x^2/(1-3x-5x^2/(1-.... (continued fraction). - Paul Barry, Apr 23 2009
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n)charpoly(A,2). - Milan Janjic, Jul 08 2010
G.f.: -1/U(0) where U(k) = x*k - 1 - x - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 28 2012
G.f.: 1/G(0) where G(k) = 1 + 2*x/(1 + 1/(1 - 2*x*(k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
G.f.: G(0)/(1+3*x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-2*x-1) - x*(2*k+1)*(2*k+3)*(2*x*k-2*x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k-x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012
From Sergei N. Gladkovskii, Feb 13 2013: (Start)
Conjecture: if the e.g.f. is E(x)= exp( exp(x) -1 + p*x) then
g.f.: (x+1-p*x)/x/(G(0)-x) - 1/x where G(k) = 2*x + 1 - p*x - x*k + x*(x*k - x - 1 + p*x)/G(k+1); (continued fraction).
So, for this sequence (p=-2), g.f.: (3*x+1)/x/( G(0)-x ) - 1/x where G(k) = 4*x + 1 - x*k + x*(x*k - 3*x - 1)/G(k+1);
(End)
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 30 2021
a(n) ~ n^(n-2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n-2)). - Vaclav Kotesovec, Jun 27 2022

Extensions

More terms from Karol A. Penson and Olivier Gérard, Oct 22 2007

A367818 Expansion of e.g.f. exp(1 - 3*x - exp(x)).

Original entry on oeis.org

1, -4, 15, -53, 178, -575, 1809, -5598, 17141, -52113, 157724, -475997, 1433429, -4311364, 12958627, -38909601, 116831426, -350844883, 1051414421, -3160120038, 9491592177, -28218244109, 86403627444, -255153772169, 722619907385, -2772952748516, 4627276967623, -17420488524253
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2023

Keywords

Crossrefs

Cf. A000587, A005494, A109747, A153732, A193683, A346738, A367819.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Exp[1 - 3 x - Exp[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] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 27}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 3*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 3*x*A(x) - x * A(x/(1 - x)) / (1 - x).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-3)^n / k!.
a(0) = 1; a(n) = -3*a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000587(k).

A367819 Expansion of e.g.f. exp(1 - 4*x - exp(x)).

Original entry on oeis.org

1, -5, 24, -111, 497, -2166, 9239, -38765, 160658, -659773, 2691205, -10922544, 44166173, -178098121, 716703848, -2879774019, 11558005677, -46348854134, 185746261419, -744036460097, 2979305960426, -11926715433881, 47735079979633, -191026723545976, 764362047956073, -3058170811731677
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2023

Keywords

Crossrefs

Cf. A000587, A045379, A109747, A153732, A193684, A346739, A367818.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[1 - 4 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 25}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 4*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 4*x*A(x) - x * A(x/(1 - x)) / (1 - x).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-4)^n / k!.
a(0) = 1; a(n) = -4*a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A000587(k).
Showing 1-3 of 3 results.

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

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