A292975 -id:A292975 - 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

A000752 Boustrophedon transform of powers of 2.

Original entry on oeis.org

1, 3, 9, 28, 93, 338, 1369, 6238, 31993, 183618, 1169229, 8187598, 62545893, 517622498, 4613366689, 44054301358, 448733127793, 4856429646978, 55650582121749, 673136951045518, 8570645832753693, 114581094529057058, 1604780986816602409, 23497612049668468078

Offset: 0

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A000079, A000734.

Column k=2 of A292975.

a000752 n = sum $ zipWith (*) (a109449_row n) a000079_list
-- Reinhard Zumkeller, Nov 03 2013

t[n_, 0] := 2^n; 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 *)
With[{nn=30},CoefficientList[Series[Exp[2x](Tan[ x]+Sec[x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 15 2018 *)

from itertools import accumulate, islice
def A000752_gen(): # generator of terms
    blist, m = tuple(), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= 2
A000752_list = list(islice(A000752_gen(),40)) # Chai Wah Wu, Jun 12 2022

E.g.f.: exp(2*x) (tan(x) + sec(x)).
a(n) = Sum_{k=0..n} A109449(n,k)*2^k. - Reinhard Zumkeller, Nov 03 2013
G.f.: E(0)*x/(1 - 2*x)/(1 - 3*x) + 1/(1 - 2*x), where E(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(x*(k+3) - 1)*(x*(k+4)  -1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014
a(n) ~ n! * exp(Pi) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

A292976 a(n) = n! * [x^n] exp(nx)(sec(x) + tan(x)).

Original entry on oeis.org

1, 2, 9, 65, 645, 8141, 124729, 2247853, 46584937, 1091386465, 28521016621, 822514469149, 25946988879053, 888784357214729, 32851731018695905, 1303291334592451037, 55235983848811714129, 2490726416399046168993, 119065442891277782378581, 6014589653389306889686941

Offset: 0

Ilya Gutkovskiy, Sep 27 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms
Index entries for sequences related to boustrophedon transform

Main diagonal of A292975.

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:= proc(n, k) option remember; `if`(k=0, b(n, 0),
      add(binomial(n, j)*A(j, k-1), j=0..n))
    end:
a:= n-> A(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 27 2017

Table[n! SeriesCoefficient[Exp[n x] (Sec[x] + Tan[x]), {x, 0, n}], {n, 0, 20}]

a(n) = A292975(n,n).

a(n) ~ (1 + sin(1)) / cos(1) * n^n. - Vaclav Kotesovec, Oct 06 2017

A307878 Expansion of e.g.f. exp(3x)(sec(x) + tan(x)).

Original entry on oeis.org

1, 4, 16, 65, 272, 1189, 5506, 27365, 147512, 868129, 5589646, 39309965, 300724652, 2489776969, 22192420786, 211923843365, 2158631018192, 23361793658209, 267706067651926, 3238110860029565, 41228900865842132, 551189774407729849, 7719762678323791066

Offset: 0

Ilya Gutkovskiy, Jun 04 2019

nonn

Boustrophedon transform of A000244 (powers of 3).

Index entries for sequences related to boustrophedon transform

Column k=3 of A292975.

Cf. A000111, A000244, A000752, A307879.

nmax = 22; CoefficientList[Series[Exp[3 x] (Sec[x] + Tan[x]), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := 3^n; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 23, 0]

from itertools import accumulate, islice
def A307878_gen(): # generator of terms
    blist, m = tuple(), 1
    yield from blist
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= 3
A307878_list = list(islice(A307878_gen(),40)) # Chai Wah Wu, Jun 12 2022

A307879 Expansion of e.g.f. exp(4x)(sec(x) + tan(x)).

Original entry on oeis.org

1, 5, 25, 126, 645, 3380, 18285, 103036, 610345, 3833540, 25714345, 185107596, 1433220045, 11932724900, 106613406405, 1019012112556, 10382757537745, 112378069315460, 1287787864054465, 15576862520435916, 198330820236011445, 2651486893149253220, 37135749401704458525

Offset: 0

Ilya Gutkovskiy, Jun 04 2019

nonn

Boustrophedon transform of A000302 (powers of 4).

Index entries for sequences related to boustrophedon transform

Column k=4 of A292975.

Cf. A000111, A000302, A000752, A307878.

nmax = 22; CoefficientList[Series[Exp[4 x] (Sec[x] + Tan[x]), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := 4^n; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 23, 0]

from itertools import accumulate, islice
def A307879_gen(): # generator of terms
    blist, m = tuple(), 1
    yield from blist
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= 4
A307879_list = list(islice(A307879_gen(),40)) # Chai Wah Wu, Jun 12 2022

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A000667 Boustrophedon transform of all-1's sequence.

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A000752 Boustrophedon transform of powers of 2.

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A292976 a(n) = n! * [x^n] exp(n*x)*(sec(x) + tan(x)).

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A307878 Expansion of e.g.f. exp(3*x)*(sec(x) + tan(x)).

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A307879 Expansion of e.g.f. exp(4*x)*(sec(x) + tan(x)).

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A292976 a(n) = n! * [x^n] exp(nx)(sec(x) + tan(x)).

A307878 Expansion of e.g.f. exp(3x)(sec(x) + tan(x)).

A307879 Expansion of e.g.f. exp(4x)(sec(x) + tan(x)).