A002084 -id:A002084 - cp's OEIS frontend

A002085 From the expansion of sinh(x) / cos(x): a(n) = odd part of A002084(n).

1, 1, 9, 39, 1141, 12721, 804309, 17113719, 1886573641, 65373260641, 11127809595009, 570506317184199, 138730500808639741, 9867549661639871761, 3247323803322747092109, 305991103023048833011479, 130958280255560469436519441, 15789929076277491342686566081

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date. [Cached copy, with permission]
J. M. Gandhi, The coefficients of sinh x/ cos x. Canad. Math. Bull. 13 1970 305-310.

Cf. A002084.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sinh[x] / Cos[x], {x, 0, 2 n + 1}] (2 n + 1)! / 2^(n + Mod[n, 2])] (* Michael Somos, Apr 10 2011 *)

{a(n) = local(A, m); if( n<0, 0, m = 2*n + 1; A = x * O(x^m) ; 2^(n + 1 - n%2) * m! * polcoeff( sinh( x/2 + A) / cos( x/2 + A), m))} /* Michael Somos, Apr 10 2011 */

Divide A002084(n) by 4^(floor((n+1)/4)).

Added more terms from b-file. - David Radcliffe, Jul 21 2025

A109449 Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 8, 6, 4, 1, 16, 25, 20, 10, 5, 1, 61, 96, 75, 40, 15, 6, 1, 272, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1, 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1, 50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1

Offset: 0

Philippe Deléham, Aug 27 2005

The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k=0..n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.
The 'signed version' of the triangle is the exponential Riordan array [sech(x) + tanh(x), x]. - Peter Luschny, Jan 24 2009
Up to signs, the matrix is self-inverse: T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 15 2013

			Triangle starts:
      1;
      1,     1;
      1,     2,     1;
      2,     3,     3,     1;
      5,     8,     6,     4,     1;
     16,    25,    20,    10,     5,    1;
     61,    96,    75,    40,    15,    6,    1;
    272,   427,   336,   175,    70,   21,    7,   1;
   1385,  2176,  1708,   896,   350,  112,   28,   8,  1;
   7936, 12465,  9792,  5124,  2016,  630,  168,  36,  9,  1;
  50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Luschny, The Swiss-Knife polynomials.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.
See also : A000182, A000964, A009739, A062161, A062272.
Cf. A153641, A162660.
Cf. A000667 (row sums), A247453.

a109449 n k = a109449_row n !! k
a109449_row n = zipWith (*)
                (a007318_row n) (reverse $ take (n + 1) a000111_list)
a109449_tabl = map a109449_row [0..]
-- Reinhard Zumkeller, Nov 02 2013

f:= func< n,x | Evaluate(BernoulliPolynomial(n+1), x) >;
A109449:= func< n,k | k eq n select 1 else 2^(2*n-2*k+1)*Binomial(n,k)*Abs(f(n-k,3/4) - f(n-k,1/4) + f(n-k,1) - f(n-k,1/2))/(n-k+1) >;
[A109449(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jul 10 2025

From Peter Luschny, Jul 10 2009, edited Jun 06 2022: (Start)
A109449 := (n,k) -> binomial(n, k)*A000111(n-k):
seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
B109449 := (n,k) -> 2^(n-k)*binomial(n, k)*abs(euler(n-k, 1/2)+euler(n-k, 1)) -`if`(n-k=0, 1, 0): seq(print(seq(B109449(n, k), k=0..n)), n=0..9);
R109449 := proc(n, k) option remember; if k = 0 then A000111(n) else R109449(n-1, k-1)*n/k fi end: seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
E109449 := proc(n) add(binomial(n, k)*euler(k)*((x+1)^(n-k)+ x^(n-k)), k=0..n) -x^n end: seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
sigma := n -> ifelse(n=0, 1, [1,1,0,-1,-1,-1,0,1][n mod 8 + 1]/2^iquo(n-1,2)-1):
L109449 := proc(n) add(add((-1)^v*binomial(k, v)*(x+v+1)^n*sigma(k), v=0..k), k=0..n) end: seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);
X109449 := n -> n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n): seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9);
(End)

lim = 10; s = CoefficientList[Series[(1 + Sin[x])/Cos[x], {x, 0, lim}], x] Table[k!, {k, 0, lim}]; Table[Binomial[n, k] s[[n - k + 1]], {n, 0, lim}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015, after Jean-François Alcover at A000111 *)
T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 27 2019 *)

A109449(n,k)=binomial(n,k)*if(n>k,2*abs(polylog(k-n,I)),1) \\ M. F. Hasler, Oct 05 2017

R = PolynomialRing(ZZ, 'x')
@CachedFunction
def skp(n, x) :
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A109449_row(n):
    x = R.gen()
    return [abs(c) for c in list(skp(n,x)-skp(n,x-1)+x^n)]
for n in (0..10) : print(A109449_row(n)) # Peter Luschny, Jul 22 2012

Sum_{k>=0} T(n, k) = A000667(n).
Sum_{k>=0} T(2n, 2k) = A000795(n).
Sum_{k>=0} T(2n, 2k+1) = A009747(n).
Sum_{k>=0} T(2n+1, 2k) = A003719(n).
Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).
Sum_{k>=0} T(n, 2k) = A062272(n).
Sum_{k>=0} T(n, 2k+1) = A062161(n).
Sum_{k>=0} (-1)^(k)*T(n, k) = A062162(n). - Johannes W. Meijer, Apr 20 2011
E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic, May 20 2007
T(n,k) = 2^(n-k)*C(n,k)*|E(n-k,1/2) + E(n-k,1)| - [n=k] where C(n,k) is the binomial coefficient, E(m,x) are the Euler polynomials and [] the Iverson bracket. - Peter Luschny, Jan 24 2009
From Reikku Kulon, Feb 26 2009: (Start)
A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).
A109449(n, n - 1) = n.
A109449(n, n) = 1.
For n > 0, k > 0: A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)
From Peter Luschny, Jul 10 2009: (Start)
Let p_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v C(k,v)*F(k)*(x+v+1)^n, where F(0)=1 and for k>0 F(k)=-1 + s_k 2^floor((k-1)/2), s_k is 0 if k mod 8 in {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute values of the coefficients of these polynomials.
Another way to express the polynomials p_n(x) is
p_n(x) = -x^n + Sum_{k=0..n} binomial(n,k)*Euler(k)((x+1)^(n-k) + x^(n-k)). (End)
From Peter Bala, Jan 26 2011: (Start)
An explicit formula for the n-th row polynomial is
x^n + i*Sum_{k=1..n}((1+i)/2)^(k-1)*Sum_{j=0..k} (-1)^j*binomial(k,j)*(x+i*j)^n, where i = sqrt(-1). This is the triangle of connection constants between the polynomial sequences {Z(n,x+1)} and {Z(n,x)}, where Z(n,x) denotes the zigzag polynomials described in A147309.
Denote the present array by M. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of A145876 - the generalized Eulerian numbers associated with the zigzag numbers. (End)
Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then
T(n,k) = [x^(n-k)] |skp{n}(x) - skp{n}(x-1) + x^n|. - Peter Luschny, Jul 22 2012
T(n,k) = A007318(n,k) * A000111(n - k), k = 0..n. - Reinhard Zumkeller, Nov 02 2013
T(n,k) = abs(A247453(n,k)). - Reinhard Zumkeller, Sep 17 2014

Edited, formula corrected, typo T(9,4)=2016 (before 2816) fixed by Peter Luschny, Jul 10 2009

A000667 Boustrophedon transform of all-1's sequence.

Original entry on oeis.org

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

A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).

Original entry on oeis.org

1, 3, 1, 5, 10, 1, 7, 35, 21, 1, 9, 84, 126, 36, 1, 11, 165, 462, 330, 55, 1, 13, 286, 1287, 1716, 715, 78, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1

Offset: 0

Ralf Stephan, Feb 06 2005

A subset of Pascal's triangle A007318.
Elements have the same parity as those of Pascal's triangle.
Matrix inverse is A104033. - Paul D. Hanna, Feb 28 2005
Row reverse of A091042. - Peter Bala, Jul 29 2013
Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. Cf. A086645. - Peter Bala, Aug 06 2013
The row polynomial P(d, x) = Sum_{k=0..d} T(d, k)*x^k, multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,3] given in A292219. - Wolfdieter Lang, Oct 12 2017

			The triangle T(n, k) begins:
n\k   0    1     2      3      4      5      6     7    8   9  10 ...
0:    1
1:    3    1
2:    5   10     1
3:    7   35    21      1
4:    9   84   126     36      1
5:   11  165   462    330     55      1
6:   13  286  1287   1716    715     78      1
7:   15  455  3003   6435   5005   1365    105     1
8:   17  680  6188  19448  24310  12376   2380   136    1
9:   19  969 11628  50388  92378  75582  27132  3876  171   1
10:  21 1330 20349 116280 293930 352716 203490 54264 5985 210   1
... reformatted and extended. - _Wolfdieter Lang_, Oct 12 2017
From _Peter Bala_, Aug 06 2013: (Start)
Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n+1)! the column generating functions begin
1st col: cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + 9*y^4/9! + ....
2nd col: 1/3!*y*cosh(sqrt(y)) = y/3! + 10*y^2/5! + 35*y^3/7! + 84*y^4/9! + ....
3rd col: 1/5!*y^2*cosh(sqrt(y)) = y^2/5! + 21*y^3/7!! + 126*y^4/9! + 462*y^5/11! + .... (End)

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.

Indranil Ghosh, Rows 0..120 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Reflected version of A091042. Cf. A086645, A103328.

Cf. A104033, A086645, A292219.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019

[Binomial(2*n+1, 2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

Flatten[Table[Binomial[2n+1,2k+1],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 19 2014 *)

create_list(binomial(2*n+1,2*k+1),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1+X*(1-Y))/((1+X*(1-Y))^2-4*X),n,x),k,y)} \\ Paul D. Hanna, Feb 28 2005

T(n,k) = binomial(2*n+1, 2*k+1);
for(n=0, 12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019

[[binomial(2*n+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f. for column k: Sum_{j=0..k+1} C(2*(k+1), 2*j)*x^j/(1-x)^(2*(k+1)). - Paul Barry, Feb 24 2005
G.f.: A(x, y) = (1 + x*(1-y))/( (1 + x*(1-y))^2 - 4*x ). - Paul D. Hanna, Feb 28 2005
Sum_{k=0..n} T(n, k)*A000364(n-k) = A002084(n). - Philippe Deléham, Aug 27 2005
E.g.f.: 1/sqrt(x)*sinh(sqrt(x)*t)*cosh(t) = t + (3 + x)*t^3/3! + (5 + 10*x + x^2)*t^5/5! + .... - Peter Bala, Jul 29 2013
T(n+2,k+2) = 2*T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k+2) + 2*T(n,k+1) - T(n,k). - Emanuele Munarini, Jul 05 2017

A003701 Expansion of e.g.f. exp(x)/cos(x).

Original entry on oeis.org

1, 1, 2, 4, 12, 36, 152, 624, 3472, 18256, 126752, 814144, 6781632, 51475776, 500231552, 4381112064, 48656756992, 482962852096, 6034272215552, 66942218896384, 929327412759552, 11394877025289216, 174008703107274752, 2336793875186479104, 38928735228629389312

Offset: 0

R. H. Hardin

nonn

Binomial transform of A000364 (with interpolated zeros). Hankel transform is A055209. - Paul Barry, Jan 12 2009

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 + 3472*x^8 + ...

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

Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)
T. Chow, Fair permutations and random k-sets, Problem 11523, Amer. Math. Monthly 117 (October 2010), 741; solution by Jim Simons, Amer. Math. Monthly 119 (November 2012), 801-803.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A009739, A062272, A062161.

Bisections are A000795 and A002084.

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)/Cos(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 14 2018

G(x):= exp(x)*sec(x): f[0]:=G(x): for n from 1 to 54 do f[n]:= diff(f[n-1],x) od: x:=0: seq(f[n], n=0..22); # Zerinvary Lajos, Apr 05 2009
# second Maple program:
b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> add(`if`(j::odd, 0, b(j, 0)*binomial(n, j)), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 12 2024

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x ] / Cos[x], {x, 0, n}]] (* Michael Somos, Jun 06 2012 *)

x='x+O('x^66); Vec(serlaplace(exp(x)/cos(x))) \\ Joerg Arndt, May 07 2013

G.f.: 1/(1-x-x^2/(1-x-4x^2/(1-x-9x^2/(1-x-16x^2.... (continued fraction). - Paul Barry, Jan 12 2009
E.g.f.: exp(x)*sec(x). - Zerinvary Lajos, Apr 05 2009
E.g.f.: 1+x/H(0); H(k)=4k+1-x+x^2*(4k+1)/((2k+1)*(4k+3)-x^2+x*(2k+1)*(4k+3)/(2k+2-x+x*(2k+2)/H(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
G.f.: 1/G(0) where G(k)= 1 - 2*x*(k+1)/(1 + 1/(1 + 2*x*(k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 20 2012
G.f.: -1/x/Q(0), where Q(k)= 1 - 1/x - (k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: (1-x)/Q(0), where Q(k)= (1-x)^2 - (1-x)^2*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ n! * ((-1)^n*exp(-Pi/2) + exp(Pi/2)) *(2/Pi)^(n+1). - Vaclav Kotesovec, Oct 08 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 - x*(2*k+2)/( x*(2*k+2) - 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/Q(k+1) ))))))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

Extended and reformatted 03/97.

A062161 Boustrophedon transform of n mod 2.

Original entry on oeis.org

0, 1, 2, 4, 12, 36, 142, 624, 3192, 18256, 116282, 814144, 6219972, 51475776, 458790022, 4381112064, 44625674352, 482962852096, 5534347077362, 66942218896384, 852334810990332, 11394877025289216, 159592488559874302, 2336793875186479104, 35703580441464231912

Offset: 0

Frank Ellermann, Jun 10 2001

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 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Cf. A000035, A062272.

a062161 n = sum $ zipWith (*) (a109449_row n) $ cycle [0,1]
-- Reinhard Zumkeller, Nov 03 2013

With[{nn=30},CoefficientList[Series[(Sec[x]+Tan[x])Sinh[x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Feb 16 2013 *)

from itertools import accumulate, islice
def A062161_gen(): # generator of terms
    blist, m = tuple(), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
A062161_list = list(islice(A062161_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Generalized algorithm of L. Seidel (1877)
def A062161_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == -1 else 0
        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
A062161_list(10) # Peter Luschny, Jun 02 2012

a(2n) = A009747(n), a(2n+1) = A002084(n).
E.g.f.: (sec(x)+tan(x))*sinh(x); a(n)=(A000667(n)-A062162(n))/2. - Paul Barry, Jan 21 2005
a(n) = Sum{k, k>=0} binomial(n, 2k+1)*A000111(n-2k-1). - Philippe Deléham, Aug 28 2005
a(n) = Sum_{k=0..n} A109449(n,k) * (k mod 2). - Reinhard Zumkeller, Nov 03 2013 [corrected by Jason Yuen, Jan 07 2025]

A104033 Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2n+1,2k+1).

Original entry on oeis.org

1, -3, 1, 25, -10, 1, -427, 175, -21, 1, 12465, -5124, 630, -36, 1, -555731, 228525, -28182, 1650, -55, 1, 35135945, -14449006, 1782495, -104676, 3575, -78, 1, -2990414715, 1229758075, -151714563, 8912475, -305305, 6825, -105, 1, 329655706465, -135565467080, 16724709820, -982532408

Offset: 0

Paul D. Hanna, Feb 28 2005

Column 0 equals signed A009843 (expansion of x/cosh(x)). Row sums form signed A000182 (expansion of tanh(x)).
The matrix logarithm is L(n,k) = -(-1)^(n-k)*A000182(n-k)*A103327(n,k), where A000182 = tangent numbers.
Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... so that 1/E(y) = 1 - 3*y/3! + 25*y^2/5! - 427*y^3/7! + .... Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. - Peter Bala, Aug 06 2013

			Rows begin:
1;
-3, 1;
25, -10, 1;
-427, 175, -21, 1;
12465, -5124, 630, -36, 1;
-555731 ,228525, -28182, 1650, -55, 1;
35135945, -14449006, 1782495, -104676, 3575, -78, 1;
-2990414715, 1229758075, -151714563, 8912475, -305305, 6825, -105, 1;
329655706465, -135565467080, 16724709820, -982532408, 33669350, -754936, 11900, -136, 1; ...
From _Peter Bala_, Aug 06 2013: (Start)
The real zeros of the row polynomials R(n,x) seem to converge to the even squares as n increases.
Polynomial |        Real zeros to 6 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 3.999986
R(10,x)    | 4.000000, 15.999978
R(15,x)    | 4.000000, 16.000000, 35.999992, 64.414273, 76.998346
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
(End)

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A000364, A103327, A009843, A000182 (unsigned row sums), A055133, A086645, A086646, A096053, A103364.

{T(n,k) = if(n=j, binomial(2*m-1,2*j-1))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))

{T(n,k) = binomial(2*n+1,2*k+1) * polcoeff(1/cosh(x+x*O(x^(2*n))), 2*n-2*k) * (2*n-2*k)!}
for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))

Column k: Sum_{j=0..n} C(2*n+1, 2*j+1) * T(j, k) = 0 (n>k), or 1 (n=k).

Row n: Sum_{j=0..n} T(n, j) * C(2*j+1, 2*k+1) = 0 (k

Sum_{k=0..n} T(n, k) * 4^k = 1 for n >= 0.

T(n, k) = (-1)^(n-k)*A000364(n-k)*A103327(n, k), where A000364 = Euler numbers.

Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A002084(n). - Philippe Deléham, Aug 27 2005

From Peter Bala, Aug 06 2013: (Start)

Generating function: 1/sqrt(x)*sinh(sqrt(x)*t)/cosh(t) = t + (-3 + x)*t^3/3! + (25 - 10*x + x^2)*t^5/5! + ....

Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(2*n+1,2*k+1)*R(k,x) with initial value R(0,x) = 1.

It appears that for arbitrary nonzero complex x we have

lim_{n -> inf} R(n,x^2)/R(n,0) = (1/(Pi/2*x))*sin(Pi/2*x).

A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,x) seem to converge to the even squares 4, 16, 36, ... as n increases. Some numerical examples are given below. Cf. A055133, A086646 and A103364.

If p = 2*n + 1 is a prime then all the entries in row n are divisible by p, apart from T(n,n) = 1. Thus the row sum is congruent to 1 modulo p.

Row sums R(n,1) = (-1)^n*A000182(n+1).

R(n,4) = 1; R(n,16) = (1/2)*( 3^(2*n+1) - 1 ) = A096053(n);

R(n,36) = (1/3)*( 5^(2*n+1) - 3^(2*n+1) + 1 );

R(n,64) = (1/4)*( 7^(2*n+1) - 5^(2*n+1) + 3^(2*n+1) - 1 ). (End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A380053 E.g.f. (exp(x) - 1)/cos(x).

Original entry on oeis.org

1, 1, 4, 7, 36, 91, 624, 2087, 18256, 76231, 814144, 4078867, 51475776, 300870571, 4381112064, 29265244847, 482962852096, 3629392540111, 66942218896384, 558956224522027, 11394877025289216, 104659828714136851, 2336793875186479104, 23414201065072302407, 568240131312188379136

Offset: 1

Paul D. Hanna, Jan 24 2025

nonn

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 7*x^4/4! + 36*x^5/5! + 91*x^6/6! + 624*x^7/7! + 2087*x^8/8! + 18256*x^9/9! + 76231*x^10/10! + ...
RELATED SERIES.
1/cos(x) = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! + 50521*x^10/10! + 2702765*x^12/12! + 199360981*x^14/14! + ... + A000364(n)*x^(2*n)/(2*n)! + ...
Let F(x) be the series reversion of e.g.f. A(x) so that A(F(x)) = x, then
F(x) = x - x^2/2! - x^3/3! + 18*x^4/4! - 86*x^5/5! - 210*x^6/6! + 8840*x^7/7! - 80080*x^8/8! - 266220*x^9/9! + ... + A380055(n)*x^n/n! + ...
where F(x) = log(1 + x*cos(F(x))).

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A002084, A000364, A380054, A380055, A380057.

{a(n) = my(X = x + x*O(x^n)); n!*polcoef( (exp(X) - 1)/cos(X), n)}
for(n=1,25,print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = (exp(x) - 1)/cos(x).
(2) A(F(x)) = x where F(x) = log(1 + x*cos(F(x))) equals the e.g.f. of A380055.
(3.a) a(2*n+1) = A002084(n) for n >= 0.
(3.b) a(2*n+1) = Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(n-k) for n >= 0.
(3.c) a(2*n) = Sum_{k=1..n} binomial(2*n, 2*k) * A000364(n-k) for n >= 1.
a(n) ~ (-1 + exp(Pi/2) + (-1 + exp(-Pi/2))*(-1)^n) * 2^(n + 3/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jan 24 2025

A301942 Expansion of e.g.f. arcsin(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 4, 44, 1016, 42384, 2908544, 306305856, 46659144832, 9760451385600, 2683733034474496, 936308392553036800, 403127865773461755904, 209562975305232836300800, 129255511221696545852424192, 93252273300325219683758915584, 77766048645578119241905858314240

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

nonn

			arcsin(x)/cos(x) = x/1! + 4*x^3/3! + 44*x^5/5! + 1016*x^7/7! + 42384*x^9/9! + ...

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012782, A296741, A302444, A302542, A302543.

nmax = 16; Table[(CoefficientList[Series[ArcSin[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x)/cos(x).

A302444 Expansion of e.g.f. arcsinh(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 2, 24, 216, 15936, -77056, 90991744, -8523712768, 2731708067840, -684815907467264, 268028469798256640, -114888252320482000896, 62022733722259702579200, -38635369828053720937463808, 28349537098304682205749968896, -23874826868622028919177351004160

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

sign

			arcsinh(x)/cos(x) = x/1! + 2*x^3/3! + 24*x^5/5! + 216*x^7/7! + 15936*x^9/9! - 77056*x^11/11! + ...

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012821, A296742, A301942, A302542, A302543.

nmax = 16; Table[(CoefficientList[Series[ArcSinh[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x)/cos(x).

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A002085 From the expansion of sinh(x) / cos(x): a(n) = odd part of A002084(n).

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A109449 Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.

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A000667 Boustrophedon transform of all-1's sequence.

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A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).

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A003701 Expansion of e.g.f. exp(x)/cos(x).

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A062161 Boustrophedon transform of n mod 2.

A104033 Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2n+1,2k+1).