A000667 -id:A000667 - cp's OEIS frontend

A009337 Expansion of e.g.f.: log(1+sin(x))/exp(x).

0, 1, -3, 7, -16, 40, -117, 411, -1720, 8384, -46561, 289595, -1991504, 14993108, -122595085, 1081616659, -10239597968, 103522029312, -1113074512289, 12681693805203, -152613117638712, 1934275341388596, -25753344727083773

Offset: 0

R. H. Hardin

This sequence starting with the 2nd term, unsigned, is the partial sums of A000667. - Gerald McGarvey, Aug 13 2004

Cf. A000667.

With[{nn=30},CoefficientList[Series[Log[1+Sin[x]]/Exp[x],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Aug 13 2012 *)

a(n) ~ (n-1)! * (-1)^(n+1) * exp(Pi/2) * 2^(n+1) / Pi^n. - Vaclav Kotesovec, Jan 23 2015

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Aug 13 2012

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]

A292975 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)(sec(x) + tan(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 9, 5, 1, 5, 16, 28, 24, 16, 1, 6, 25, 65, 93, 77, 61, 1, 7, 36, 126, 272, 338, 294, 272, 1, 8, 49, 217, 645, 1189, 1369, 1309, 1385, 1, 9, 64, 344, 1320, 3380, 5506, 6238, 6664, 7936, 1, 10, 81, 513, 2429, 8141, 18285, 27365, 31993, 38177, 50521

Offset: 0

Ilya Gutkovskiy, Sep 27 2017

A(n,k) is the k-th binomial transform of A000111 evaluated at n.

Also column k is the boustrophedon transform of powers of k.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k + 1)^2*x^2/2! + (k^3 + 3*k^2 + 3*k + 2)*x^3/3! + (k^4 + 4*k^3 + 6*k^2 + 8*k + 5)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   1,   4,    9,    16,    25,    36,  ...
   2,   9,   28,    65,   126,   217,  ...
   5,  24,   93,   272,   645,  1320,  ...
  16,  77,  338,  1189,  3380,  8141,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
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.
Index entries for sequences related to boustrophedon transform

Columns k=0..2 give A000111, A000667, A000752.

Main diagonal gives A292976.

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:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017

Table[Function[k, n! SeriesCoefficient[Exp[k x] (Sec[x] + Tan[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(k*x)*(sec(x) + tan(x)).

A062704 Di-Boustrophedon transform of all 1's sequence: Fill in an array by diagonals alternating in the 'up' and 'down' directions. Each diagonal starts with a 1. When going in the 'up' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the row the new element is in. When going in the 'down' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the column the new element is in. The final element of the n-th diagonal is a(n).

Original entry on oeis.org

1, 2, 5, 13, 40, 145, 616, 3017, 16752, 103973, 713040, 5352729, 43645848, 384059537, 3626960272, 36585357429, 392545057280, 4463791225145, 53622168102640, 678508544425721, 9020035443775264, 125684948107190045, 1831698736650660952, 27866044704218390113

Offset: 1

Floor van Lamoen, Jul 11 2001

			The array begins:
   1   2   1  13   1
   1   3  10  14
   5   6  25
   1  34
  40

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000667, A059216, A063179.

T:= proc(n, k) option remember;
      if n<1 or k<1 then 0
    elif n=1 and irem(k, 2)=1 or k=1 and irem(n, 2)=0 then 1
    elif irem(n+k, 2)=0 then T(n-1, k+1)+T(n-1, k)+T(n-2, k)
                        else T(n+1, k-1)+T(n, k-1)+T(n, k-2)
      fi
    end:
a:= n-> `if`(irem (n, 2)=0, T(1, n), T(n, 1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 08 2011

T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0
     , n == 1 && Mod[k, 2] == 1 || k == 1 && Mod[n, 2] == 0, 1
     , Mod[n + k, 2] == 0, T[n - 1, k + 1] + T[n - 1, k] + T[n - 2, k]
     , True,               T[n + 1, k - 1] + T[n, k - 1] + T[n, k - 2]];
a[n_] := If[Mod [n, 2] == 0, T[1, n], T[n, 1]];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

More terms from Alois P. Heinz, Feb 08 2011

A059512 For n>=2, the number of (s(0), s(1), ..., s(n-1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n-1, s(0) = 2, s(n-1) = 2.

Original entry on oeis.org

0, 1, 1, 3, 7, 18, 46, 119, 309, 805, 2101, 5490, 14356, 37557, 98281, 257231, 673323, 1762594, 4614226, 12079707, 31624285, 82792161, 216750601, 567457058, 1485616392, 3889385353, 10182528721, 26658183099, 69791991919

Offset: 0

Floor van Lamoen, Jan 21 2001

Substituting x(1-x)/(1-2x) into x/(1-x^2) yields g.f. of sequence.

Cf. A000667, A059216, A059219, A059502, A027994.

a(1-2n)=A005207(2n), a(-2n)=A056014(2n+1).

CoefficientList[Series[x(1-x)(1-2x)/((1-x-x^2)(1-3x+x^2)), {x,0,30}], x]  (* Harvey P. Dale, Apr 23 2011 *)

a(n)=(fibonacci(2*n-1)+fibonacci(n-2))/2

a(n) = 2a(n-1) + Sum{mA000045).
G.f.: x(1-x)(1-2x)/((1-x-x^2)(1-3x+x^2)).
a(n+1)=sum{k=0..floor(n/2), C(n,2k)*F(2k+1)}. [From Paul Barry, Oct 14 2009]

A063179 Di-Boustrophedon transform of (1,0,0,0,...): Fill in an array by diagonals alternating in the 'up' and 'down' directions. The n-th diagonal starts with the n-th element of (1,0,0,0,...). When going in the 'up' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the row the new element is in. When going in the 'down' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the column the new element is in. The final element of the n-th diagonal is a(n).

Original entry on oeis.org

1, 1, 2, 4, 12, 42, 178, 870, 4830, 29976, 205572, 1543210, 12583242, 110725638, 1045664646, 10547679660, 113172039256, 1286925785286, 15459448549274, 195616259182162, 2600506074185090, 36235386548738016, 528084808585261568, 8033872923106040478

Offset: 1

Floor van Lamoen, Jul 09 2001

			Array begins:
   1  1  0  4  0 42 ...
   0  1  3  4 38 ...
   2  2  7 31 ...
   0 10 22 ...
  12 12 ...
   0 ...

Cf. A000667, A059216, A062704, A063180, A063181.

More terms from Sean A. Irvine, Apr 21 2023

A227862 A boustrophedon triangle.

Original entry on oeis.org

1, 1, 2, 4, 3, 1, 1, 5, 8, 9, 24, 23, 18, 10, 1, 1, 25, 48, 66, 76, 77, 294, 293, 268, 220, 154, 78, 1, 1, 295, 588, 856, 1076, 1230, 1308, 1309, 6664, 6663, 6368, 5780, 4924, 3848, 2618, 1310, 1, 1, 6665, 13328, 19696, 25476, 30400, 34248, 36866, 38176, 38177

Offset: 0

Reinhard Zumkeller, Nov 01 2013

T(n, n * (n mod 2)) = A000667(n).

			First nine rows:
.  0:                                    1
.  1:                               1   ->  2
.  2:                           4   <-   3  <-  1
.  3:                       1 ->    5 ->   8   ->   9
.  4:                   24  <-  23  <-  18  <-  10  <-  1
.  5:              1  ->  25  ->  48  ->   66 ->   76  ->  77
.  6:          294 <-  293 <-  268 <-  220 <-  154  <-  78   <-  1
.  7:      1  ->  295 ->  588 ->  856 -> 1076 -> 1230 -> 1308 -> 1309
.  8:  6664 <- 6663 <- 6368 <- 5780 <- 4924 <- 3848 <- 2618 <- 1310  <- 1 .

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Luschny, An old operation on sequences: the Seidel transform.
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]
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A008280.

a227862 n k = a227862_tabl !! n !! k
a227862_row n = a227862_tabl !! n
a227862_tabl = map snd $ iterate ox (False, [1]) where
   ox (turn, xs) = (not turn, if turn then reverse ys else ys)
      where ys = scanl (+) 1 (if turn then reverse xs else xs)

T[0, 0] = 1; T[n_?OddQ, 0] = 1; T[n_?EvenQ, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = If[OddQ[n], T[n, k - 1] + T[n - 1, k - 1], T[n, k + 1] + T[n - 1, k]]; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 23 2019 *)

A059513 Variation of Boustrophedon transform applied to 1,1,1,1,... Fill an array by diagonals, in alternating directions. The first entry is 1 each time. For the next element of a diagonal, add to the previous element the elements of the row and the column the new element is in. The final element of each diagonal gives a(n).

Original entry on oeis.org

1, 2, 6, 23, 116, 736, 5659, 50796, 521040, 6006587, 76874524, 1081439062, 16586149365, 275442822510, 4924040788654

Offset: 1

Floor van Lamoen, Jan 23 2001

			The array begins
1 ....2 ...1 ..23 ..1 ...
1 ....4 ..19 ..48 ...
6 ...13 ..87 ...
1 ..107 ...
116 ...
1 ...

Cf. A000667, A035002, A059216, A059219, A059575, A059574.

A059574 The array described in A059513 read by antidiagonals in the 'up' direction.

Original entry on oeis.org

1, 1, 2, 6, 4, 1, 1, 13, 19, 23, 116, 107, 87, 48, 1, 1, 243, 458, 635, 708, 736, 5659, 5533, 5163, 4239, 2967, 1517, 1, 1, 11562, 22824, 33291, 41772, 47733, 50031, 50796, 521040, 515254, 497789, 452016, 385422, 301161, 204598, 103125, 1

Offset: 1

Floor van Lamoen, Jan 23 2001

Cf. A000667, A035002, A059216, A059219, A059575.

A292756 Expansion of e.g.f. exp(x)*(tan(x)+sec(x))^2.

Original entry on oeis.org

1, 3, 9, 29, 105, 433, 2029, 10709, 63025, 409713, 2917749, 22599389, 189200745, 1702839193, 16397919469, 168248900069, 1832540103265, 21118133619873, 256725609081189, 3283461599832749, 44074814323890585, 619564244562907753, 9102144892897546909, 139495284063955397429

Offset: 0

N. J. A. Sloane, Sep 26 2017

nonn

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.

Cf. A000667.

nmax = 20; CoefficientList[Series[Exp[x]*(Tan[x]+Sec[x])^2, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 02 2019 *)

a(n) ~ 2^(n + 9/2) * n^(n + 3/2) / (Pi^(n + 3/2) * exp(n - Pi/2)). - Vaclav Kotesovec, Jun 02 2019

cp's OEIS Frontend

A009337 Expansion of e.g.f.: log(1+sin(x))/exp(x).

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A062161 Boustrophedon transform of n mod 2.

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A292975 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*(sec(x) + tan(x)).

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A059512 For n>=2, the number of (s(0), s(1), ..., s(n-1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n-1, s(0) = 2, s(n-1) = 2.

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A227862 A boustrophedon triangle.

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A059574 The array described in A059513 read by antidiagonals in the 'up' direction.

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A292975 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)(sec(x) + tan(x)).