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A173253 Partial sums of A000111. 1
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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

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Last modified April 8 11:56 EDT 2020. Contains 333314 sequences. (Running on oeis4.)