A173253 Partial sums of A000111.

1, 2, 3, 5, 10, 26, 87, 359, 1744, 9680, 60201, 413993, 3116758, 25485014, 224845995, 2128603307, 21520115452, 231385458428, 2636265133869, 31725150246701, 402096338484226, 5353594391608322, 74702468784746223, 1090126355291598575, 16604660518848685480

Offset

0,2

Comments

Partial sums of Euler or up/down numbers. Partial sums of expansion of sec x + tan x. Partial sums of number of alternating permutations on n letters.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..485

Formula

a(n) = SUM[i=0..n] A000111(i) = SUM[i=0..n] (2^i|E(i,1/2)+E(i,1)| where E(n,x) are the Euler polynomials).

G.f.: (1 + x/Q(0))/(1-x),m=+4,u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013

G.f.: 1/(1-x) + T(0)*x/(1-x)^2, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2013

a(n) ~ 2^(n+2)*n!/Pi^(n+1). - Vaclav Kotesovec, Oct 27 2016

Example

a(22) = 1 + 1 + 1 + 2 + 5 + 16 + 61 + 272 + 1385 + 7936 + 50521 + 353792 + 2702765 + 22368256 + 199360981 + 1903757312 + 19391512145 + 209865342976 + 2404879675441 + 29088885112832 + 370371188237525 + 4951498053124096 + 69348874393137901.

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, a(n-1))+ b(n, 0)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 27 2017

Mathematica

With[{nn=30}, Accumulate[CoefficientList[Series[Sec[x]+Tan[x], {x, 0, nn}], x] Range[0, nn]!]] (* Harvey P. Dale, Feb 26 2012 *)

Prog

(Python)
from itertools import accumulate
def A173253(n):
    if n<=1:
        return n+1
    c, blist = 2, (0, 1)
    for _ in range(n-1):
        c += (blist := tuple(accumulate(reversed(blist), initial=0)))[-1]
    return c # Chai Wah Wu, Apr 16 2023

Crossrefs

Cf. A000111, A000364, A000182, A008280, A008281, A008282, A010094, A059720, A008970, A109449, A162170.

Sequence in context: A260304 A259438 A135961 * A011828 A011829 A011830

Adjacent sequences: A173250 A173251 A173252 * A173254 A173255 A173256

Keyword

nonn

Author

Jonathan Vos Post, Feb 14 2010

Status

approved