A000111 -id:A000111 - cp's OEIS frontend

A185425 Bisection of A185424. Numerators of even-indexed generalized Bernoulli numbers associated with the zigzag numbers A000111.

1, 1, 19, 253, 3319, 222557, 422152729, 59833795, 439264083023, 76632373664299, 4432283799315809, 317829581058418253, 1297298660169509319229, 696911453333335463719, 28877308885785768720478751, 157040990105362922778773747849

Offset: 0

Peter Bala, Feb 18 2011

Let E(t) = sec(t)+tan(t) denote the generating function for the zigzag numbers A000111. The zigzag Bernoulli numbers, denoted ZB(n), are defined by means of the generating function log E(t)/(E(t)-1) = Sum_{n>=0} ZB(n)*t^n/n!. See formula (1).

The present sequence lists the numerators of ZB(2*n) for n>=0.

G. C. Greubel, Table of n, a(n) for n = 0..235

Cf. A000111, A027641, A000367, A185424.

Sequence of denominators is A002445.

#A185425
a := n - > (-1)^n*add (binomial(2*n,2*k)/(2*k+1)* bernoulli(2*n-2*k)*
euler(2*k), k = 0..n):
seq(numer(a(n)), n = 0..20);

Numerator[Table[(-1)^n*Sum[Binomial[2*n, 2*k]*BernoulliB[2*(n - k)]* EulerE[2*k]/(2*k + 1), {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Jul 06 2017 *)

(1)... 1/2*log(sec(t)+tan(t))*(1+sin(t)+cos(t))/(1+sin(t)-cos(t))
= Sum_{n >= 0} ZB(2*n)*t^(2*n)/(2*n)!
= 1 + (1/6)*t^2/2! + (19/30)*t^4/4! + (253/42)*t^6/6! + ....
(2)... ZB(2*n) = (-1)^n*Sum_{k = 0..n} binomial(2*n,2*k)/(2*k+1)* Bernoulli(2*n-2*k)*Euler(2*k).
(3)... a(n) = numerator(ZB(2*n)).

A091303 Third diagonal of A008970 (after A000111 and A000708).

Original entry on oeis.org

1, 14, 118, 926, 7311, 59982, 517496, 4717222, 45484301, 463683670, 4991117034, 56630860638, 676055834971, 8475491147678, 111376106002732, 1531384454297174, 21993657178539321, 329402210042157606

Offset: 1

N. J. A. Sloane, Feb 21 2004

More terms from David Wasserman, Feb 28 2006

A111413 a(n) = f(f(n+1))-f(f(n)), where f(m) = Euler(m) = A000111(m).

Original entry on oeis.org

0, 0, 0, 15, 19391512129, 703237958001393736999896827714634659411015090272684227831001142371615151

Offset: 0

N. J. A. Sloane, Nov 12 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..6

Cf. A000111.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> (f-> f(f(n+1))-f(f(n)))(k-> b(k, 0)):
seq(a(n), n=0..5);  # Alois P. Heinz, Aug 17 2021

$RecursionLimit = Infinity;
t[n_, 0] := Boole[n == 0];
t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];
f[n_] := t[n, n];
a[n_] := f[f[n+1]] - f[f[n]];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Apr 25 2022 *)

Definition corrected by N. J. A. Sloane, Feb 14 2018

A141458 a(n) = A000111(2n) + A000111(2n+1).

Original entry on oeis.org

2, 3, 21, 333, 9321, 404313, 25071021, 2103118293, 229256855121, 31493764788273, 5321869241361621, 1084772760899990253, 262436014353765070521, 74338674113237083780233, 24371443829213227706941821, 9155506650374192400724494213, 3906927095300068860534174415521

Offset: 0

Paul Curtz, Aug 08 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A000111.

A000111 := proc(n) if n = 0 then 1; else 2^n*abs(euler(n, 1/2)+euler(n, 1) ); end if; end proc:
A141458 := proc(n) A000111(2*n)+A000111(2*n+1) ; end proc: # R. J. Mathar, Jul 07 2011

terms = 20;
CoefficientList[Sec[x] + Sec[x]^2 + O[x]^(2 terms - 1), x^2] Range[0, 2 terms - 2, 2]!  (* Jean-François Alcover, Nov 19 2020 *)

from itertools import accumulate, islice
def A141458_gen(): # generator of terms
    yield 2
    blist = (0,1)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1] + (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A141458_list = list(islice(A141458_gen(),30)) # Chai Wah Wu, Apr 17 2023

a(2n) == 21 (mod 100), n>0.

a(n) = (2n)! * [x^(2n)] sec(x) + sec(x)^2. - Ira M. Gessel, Apr 18 2013

A180419 a(n) = (A000111(2p) - 1)/(2p), where p = A000040(n) = prime(n).

Original entry on oeis.org

1, 10, 5052, 14240070, 3152221563324450, 157195096511273995860, 2374214683408467590063771983920, 618146855974818638210995488847340730, 144946467754033586465978879886385830380958862710

Offset: 1

Vladimir Shevelev, Sep 03 2010

nonn

My comment in A000111 concerning A000111(2*p) mod (2*p) says that all entries are integers.

Cf. A180418, A000111, A180417.

t = Range[0, 60]! CoefficientList[ Series[ Sec@x + Tan@x, {x, 0, 60}], x]; f[n_] := (Rest[t][[2 Prime@n]] - 1)/(2 Prime@n); Array[f, 9] (* Robert G. Wilson v, Sep 04 2010 *)

from sympy import euler, prime
def A180419(n): return (1-euler(2*(p:=prime(n))))//p>>1 if n > 1 else 1 # Chai Wah Wu, Apr 18 2023

a(6) onwards from Robert G. Wilson v, Sep 04 2010

Definition rephrased by R. J. Mathar, Sep 29 2010

A231895 a(n) = 2*A000111(n+1) + A000111(n).

Original entry on oeis.org

3, 3, 5, 12, 37, 138, 605, 3042, 17257, 108978, 758105, 5759322, 47439277, 421090218, 4006875605, 40686781602, 439122198097, 5019624693858, 60582649901105, 769831261587882, 10273367294485717, 143649246839399898, 2100196647406842605, 32044492213621026162, 509357494543973054137

Offset: 0

N. J. A. Sloane, Nov 18 2013

nonn

It is clear from the Berry et al. article that they intended to consider 2*A000111(n+1) - A000111(n) (which is A104854), not 2*A000111(n+1) + A000111(n).

D. Berry, J. Broom, D. Dixon, and A. Flaherty, Umbral Calculus and the Boustrophedon Transform, 2013.

Cf. A000111, A104854.

from itertools import accumulate, islice
def A231895_gen(): # generator of terms
    yield 3
    blist = (0,1)
    while True:
        yield blist[-1]+2*(blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A231895_list = list(islice(A231895_gen(),40)) # Chai Wah Wu, Jun 14 2022

E.g.f.: 1 + (sec(x) + tan(x) + 1)*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Jun 11 2015

A253671 a(n) = floor(A000111(n)/A000111(n-1)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40

Offset: 1

Paul Curtz, Jan 08 2015, with the help of Jean-François Alcover

1, 2, 3, 4, ... first appear at n = 1, 3, 5, 7, 8, 10, 11, 13, ... . a(500) = 318.
Numbers appearing only once: interleave 4+7*n, 6+7*n, 9+7*n = 4, 6, 9, 11, 13, 16, ... .
This is a nondecreasing sequence.
The ratio a(n)/n asymptotically tends to 7/11 = 0.6363... - Jean-François Alcover, Jul 21 2015

			Floor of 1/1, 1/1, 2/1, 5/2, 16/5, 61/16, ... .
1=1*1+0, 1=1*1+0, 2=2*1+0, 5=2*2+1, 16=3*5+1, 61=3*16+13, 272=4*61+28, ... .

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A000111, A010727, A017005, A017029, A017053.

max = 500; ee = Table[2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, max}]; A000111 = Table[Differences[ee, n] // First // Abs, {n, 0, max}]; Table[Quotient[A000111[[n + 1]], A000111[[n]]], {n, 1, max}] (* Jean-François Alcover, Jan 08 2015 *)

Vec(x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1)/((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 22 2015

# requires python 3.2 or higher
from itertools import accumulate
A253671_list, blist, l1, l2 = [1], [1], 1, 1
for n in range(10**2):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    l2, l1 = l1, sum(blist)
    A253671_list.append(l1//l2) # Chai Wah Wu, Jan 29 2015

a(n+2) = a(n+1) + (0, 1, 0, followed by a sequence of period 11: repeat 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1).
a(n+12) = a(n+1) + (6, 7, 6, followed by 7's = A010727).
a(n) = a(n-1) + a(n-11) - a(n-12) for n>15. - Colin Barker, Jan 22 2015
G.f.: x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1) / ((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). - Colin Barker, Jan 22 2015

A340315 Decimal expansion of sum of reciprocals of A000111(n) (where A000111(n) is the n-th Euler or up/down number).

Original entry on oeis.org

3, 7, 8, 3, 4, 4, 0, 9, 8, 3, 5, 0, 0, 7, 9, 7, 6, 6, 6, 6, 8, 3, 6, 8, 1, 6, 3, 9, 2, 2, 7, 9, 8, 9, 1, 5, 6, 1, 0, 7, 5, 0, 8, 5, 8, 9, 8, 6, 0, 1, 1, 2, 7, 4, 3, 1, 9, 8, 1, 9, 8, 0, 6, 3, 6, 6, 9, 1, 0, 7, 1, 0, 1, 1, 2, 5, 2, 3, 2, 2, 7, 6, 3, 6, 3, 4, 2, 2, 0, 6, 6, 9, 8, 1, 1, 9, 3, 7, 1, 8, 5, 6, 3, 9

Offset: 1

Marco Ripà, Jan 04 2021

The series 1/1 + 1/1 + 1/1 + 1/2 + 1/5 + 1/16 + 1/61 + 1/272 + 1/1385 + 1/7936 + ... converges to 3.783440983500797666683...

			3.783440983500797666683...

Cf. A000111, A001250, A173253.

f(n)=if(n, 2*abs(polylog(-n, I)), 1); \\ A000111
suminf(n=0, 1/f(n)) \\ Michel Marcus, Jan 04 2021

Equals Sum_{k>=0} 1/A000111(k).

More digits from Alois P. Heinz, Jan 06 2021

A373425 a(n) = A000111(n) * A000262(n). Row sums of A373426.

Original entry on oeis.org

1, 1, 3, 26, 365, 8016, 247111, 10236176, 546178905, 36478244608, 2977762858411, 291550484700672, 33703918027674245, 4540228104291094528, 704744561517173519343, 124836607292749756516352, 25023470823661358817690545, 5634174369285939855014166528, 1415592664236058550974684119763

Offset: 0

Peter Luschny, Jun 07 2024

nonn

Cf. A000111, A000262, A373426.

A007316 Reversion of g.f. for Euler numbers A000111(n-1).

Original entry on oeis.org

1, -1, 1, -2, 3, -9, 9, -71, -96, -1325, -6843, -54922, -417975, -3586117, -32531983, -316599861, -3274076017, -35914014266, -416386323306, -5088908019824, -65392831090975, -881473287321301, -12437647407521019, -183345613125389337

Offset: 1

N. J. A. Sloane, Mira Bernstein

sign

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for reversions of series

Cf. A000111.

# Using function CompInv from A357588.
CompInv(24, n -> if n=1 then 1 else 2^(n-1)*abs(euler(n-1, 1/2) + euler(n-1, 1)) fi); # Peter Luschny, Oct 05 2022

Extended with signs by Christian G. Bower, Feb 15 1999

Signs and a(23) and a(24) corrected by Sean A. Irvine, Dec 08 2017

cp's OEIS Frontend

A185425 Bisection of A185424. Numerators of even-indexed generalized Bernoulli numbers associated with the zigzag numbers A000111.

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A091303 Third diagonal of A008970 (after A000111 and A000708).

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A111413 a(n) = f(f(n+1))-f(f(n)), where f(m) = Euler(m) = A000111(m).

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A141458 a(n) = A000111(2n) + A000111(2n+1).

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A180419 a(n) = (A000111(2*p) - 1)/(2*p), where p = A000040(n) = prime(n).

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A231895 a(n) = 2*A000111(n+1) + A000111(n).

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A253671 a(n) = floor(A000111(n)/A000111(n-1)).

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A340315 Decimal expansion of sum of reciprocals of A000111(n) (where A000111(n) is the n-th Euler or up/down number).

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A180419 a(n) = (A000111(2p) - 1)/(2p), where p = A000040(n) = prime(n).