A009337 -id:A009337 - cp's OEIS frontend

A000667 Boustrophedon transform of all-1's sequence.

1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509

Offset: 0

N. J. A. Sloane and Simon Plouffe

Fill in a triangle, like Pascal's triangle, beginning each row with a 1 and filling in rows alternately right to left and left to right.

Row sums of triangle A109449. - Reinhard Zumkeller, Nov 04 2013

			...............1..............
............1..->..2..........
.........4..<-.3...<-..1......
......1..->.5..->..8...->..9..

Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Absolute value of pairwise sums of A009337.
Column k=1 of A292975.
Cf. A000111, A000795, A009747, A002084, A003719, A109449, A227862.

a000667 n = if x == 1 then last xs else x
            where xs@(x:_) = a227862_row n
-- Reinhard Zumkeller, Nov 01 2013

With[{nn=30},CoefficientList[Series[Exp[x](Tan[x]+Sec[x]),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 28 2011 *)
t[, 0] = 1; t[n, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];
a[n_] := t[n, n];
Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ Joerg Arndt, Jul 30 2016

from itertools import islice, accumulate
def A000667_gen(): # generator of terms
    blist = tuple()
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=1)))[-1]
A000667_list = list(islice(A000667_gen(),20)) # Chai Wah Wu, Jun 11 2022

# Algorithm of L. Seidel (1877)
def A000667_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # print [A[z] for z in (-i//2..i//2)]
        R.append(A[e*i//2])
    return R
A000667_list(10)  # Peter Luschny, Jun 02 2012

E.g.f.: exp(x) * (tan(x) + sec(x)).
Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi; lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1 + 1/(n-1). - Gerald McGarvey, Aug 13 2004
a(n) = Sum_{k=0..n} binomial(n, k)*A000111(n-k). a(2*n) = A000795(n) + A009747(n), a(2*n+1) = A002084(n) + A003719(n). - Philippe Deléham, Aug 28 2005
a(n) = A227862(n, n * (n mod 2)). - Reinhard Zumkeller, Nov 01 2013
G.f.: E(0)*x/(1-x)/(1-2*x) + 1/(1-x), where E(k) = 1 - x^2*(k + 1)*(k + 2)/(x^2*(k + 1)*(k + 2) - 2*(x*(k + 2) - 1)*(x*(k + 3) - 1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014
a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

A296436 Expansion of e.g.f. log(1 + arcsin(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 3, 0, 28, -85, 1029, -6440, 79136, -724305, 9982005, -118974856, 1858582100, -27126378357, 478338929509, -8227405849840, 162502213354272, -3209170996757057, 70409595412300877, -1566861832498793248, 37885426233247176772, -936732798302547171509, 24780850678372964078189

Offset: 0

Ilya Gutkovskiy, Dec 12 2017

sign

			E.g.f.: A(x) = x/1! + x^2/2! + 3*x^3/3! + 28*x^5/5! - 85*x^6/6! + 1029*x^7/7! - 6440*x^8/8! + ...

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A009334, A009337, A009353, A189815, A291482, A296437.

a:=series(log(1+arcsin(x))*exp(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!

my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + asin(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2)))*exp(x), where i is the imaginary unit.

a(n) ~ -(-1)^n * sqrt(Pi) * 2^((n + 1)/2) * n^(n - 1/2) / (exp(n + sin(1)) * (1 - cos(2))^(n/2)). - Vaclav Kotesovec, Dec 21 2017

A297213 Expansion of e.g.f. log(1 + arctanh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 10, -40, 213, -1383, 11002, -100616, 1062625, -12508067, 164543938, -2368224032, 37311284645, -634900302775, 11658800863330, -229004281334768, 4804124787023265, -106986109080667043, 2524701174424967130, -62860054802079553016, 1648303843512405478485

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arctanh(x))*exp(-x) = x/1! - 3*x^2/2! + 10*x^3/3! - 40*x^4/4! + 213*x^5/5! - 1383*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..432

Cf. A009337, A009356, A009374, A009393, A202139, A296439, A297209, A297210, A297211.

S:= series(log(1+arctanh(x))*exp(-x),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Jul 09 2018

nmax = 21; CoefficientList[Series[Log[1 + ArcTanh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[1 + (Log[1 + x] - Log[1 - x])/2] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A297209 Expansion of e.g.f. log(1 + arcsin(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 9, -32, 148, -853, 6027, -49576, 470624, -5005137, 59454923, -774282632, 11035740844, -169997137269, 2826070412955, -50256453936368, 954657085889760, -19247168446169665, 411277539407862707, -9269937746437524256, 220085825544691181500, -5483977295221312280757

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arcsin(x))*exp(-x) = x/1! - 3*x^2/2! + 9*x^3/3! - 32*x^4/4! + 148*x^5/5! - 853*x^6/6! + ...

Cf. A009337, A009356, A009374, A009393, A189815, A296436, A297210, A297211, A297213.

a:=series(log(1+arcsin(x))*exp(-x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

x='x+O('x^99); concat([0], Vec(serlaplace(exp(-x)*log(1+asin(x))))) \\ Altug Alkan, Dec 28 2017

A297210 Expansion of e.g.f. log(1 + arcsinh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 7, -16, 48, -213, 1027, -4856, 32512, -309377, 2527963, -16805072, 179877332, -2916171997, 32511289795, -227822369168, 3575741575680, -98643332014049, 1352701143217491, -6534261348983096, 168508582018012980, -9094443640555413357, 143341194607564099595

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arcsinh(x))*exp(-x) = x/1! - 3*x^2/2! + 7*x^3/3! - 16*x^4/4! + 48*x^5/5! - 213*x^6/6! + ...

Cf. A009337, A009356, A009374, A009393, A296435, A296437, A297209, A297211, A297213.

a:=series(log(1+arcsinh(x))*exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Log[1 + ArcSinh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A297211 Expansion of e.g.f. log(1 + arctan(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 6, -8, 13, -103, 462, 824, -8239, -147747, 1233518, 12148288, -127674419, -2090702391, 24495009510, 410685350032, -5514147250815, -111860639828131, 1673006899192118, 37306857729115304, -619246417449233555, -15476404474443728487, 281907759055194714206

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arctan(x))*exp(-x) = x/1! - 3*x^2/2! + 6*x^3/3! - 8*x^4/4! + 13*x^5/5! - 103*x^6/6! + ...

Cf. A009337, A009356, A009374, A009393, A110708, A296438, A297209, A297210, A297213.

a:=series(log(1+arctan(x))*exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Log[1 + ArcTan[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + (I/2) (Log[1 - I x] - Log[1 + I x])] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

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A000667 Boustrophedon transform of all-1's sequence.

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A296436 Expansion of e.g.f. log(1 + arcsin(x))*exp(x).

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A297213 Expansion of e.g.f. log(1 + arctanh(x))*exp(-x).

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A297209 Expansion of e.g.f. log(1 + arcsin(x))*exp(-x).

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A297210 Expansion of e.g.f. log(1 + arcsinh(x))*exp(-x).

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A297211 Expansion of e.g.f. log(1 + arctan(x))*exp(-x).

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Maple

Mathematica