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.

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A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2*k - 1)^n / (2^k * k!).

Original entry on oeis.org

1, 0, 2, 4, 20, 96, 552, 3536, 25104, 194816, 1637408, 14792768, 142761280, 1464117760, 15886137984, 181667507456, 2182268117248, 27456279388160, 360872502280704, 4943580063237120, 70437638474568704, 1041911242274562048, 15972832382065977344, 253388070573020401664
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Links

Crossrefs

Cf. A000296, A004211, A007405, A124311, A166922, A217203, A337039, A337040, A337041, A337042, A337043.

Programs

  • Maple
    E:= exp((exp(2*x)-1)/2-x):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Aug 26 2020
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[(Exp[2 x] - 1)/2 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 2^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 23}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 2*x + x*A(x/(1 - 2*x))) / (1 - x - 2*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 2*j*x/(1 + x)).
E.g.f.: exp((exp(2*x) - 1) / 2 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 2^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004211(k).
a(n) ~ 2^(n - 1/2) * n^(n - 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

A217202 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.

Original entry on oeis.org

0, 1, 2, 7, 2, 28, 16, 131, 118, 16, 690, 892, 272, 4033, 7060, 3468, 272, 25864, 58608, 41088, 7936, 180265, 510812, 479772, 156176, 7936, 1354458, 4675912, 5635224, 2665184, 353792, 10898823, 44918110, 67238764, 42832648, 9972704, 353792, 93407828, 452104928
Offset: 1

Views

Author

N. J. A. Sloane, Sep 27 2012

Keywords

Comments

See Ma (2012) for precise definition (cf. Proposition 6).

Examples

			Triangle begins:
    0;
    1;
    2;
    7,   2;
   28,  16;
  131, 118,  16;
  690, 892, 272;
  ...

Links

Crossrefs

First column is A217203.

Programs

  • Mathematica
    V[0][, ] = 1; V[1][, ] = 0; V[2][, x] := x; V[3][, x] := 2x;
V[n_][q_, x_] := V[n][q, x] = (n-1) q V[n-1][q, x] + 2q(1-q) D[V[n-1][q, x], q] + 2x (1-q) D[V[n-1][q, x], x] + (n-1) x V[n-2][q, x] // Simplify;
Table[If[n==1, {0}, CoefficientList[V[n][q, x] /. x -> 1, q]], {n, 1, 13}] // Flatten (* Jean-François Alcover, Sep 23 2018 *)
  • PARI
    tabf(m) = {P = x; M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); Q = 2*x; M = subst(Q, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); for (n=3, m, newP = n*q*Q + 2*q*(1-q)*deriv(Q,q) + 2*x*(1-q)*deriv(Q,x) + n*x*P; M = subst(newP, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = Q; Q = newP;);} \\ Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013
Showing 1-2 of 2 results.

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

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