A109449 -id:A109449 - cp's OEIS frontend

A000736 Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...

1, 2, 4, 10, 32, 120, 513, 2455, 13040, 76440, 492231, 3465163, 26530503, 219754535, 1959181266, 18710532565, 190588702776, 2062664376064, 23636408157551, 285900639990875, 3640199365715769, 48665876423760247

Offset: 0

N. J. A. Sloane, Simon Plouffe

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

Cf. A000108, A000753, A109449.

a000736 n = sum $ zipWith (*) (a109449_row n) (1 : a000108_list)
-- Reinhard Zumkeller, Nov 05 2013

egf := (sec(x/2)+tan(x/2))*(exp(x)*((x-1/2)*BesselI(0,x)-x*BesselI(1,x))+3/2);
s := n -> 2^n*n!*coeff(series(egf,x,n+2),x,n); seq(s(n), n=0..22); # Peter Luschny, Oct 30 2014, after Sergei N. Gladkovskii

CoefficientList[Series[1/2*(3 + E^(2*x)*((4*x-1)*BesselI[0, 2*x] - 4*x*BesselI[1, 2*x]))*(Sec[x] + Tan[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 30 2014, after Peter Luschny *)
t[n_, 0] := If[n == 0, 1, CatalanNumber[n - 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 *)

from itertools import accumulate, count, islice
def A000736_gen(): # generator of terms
    yield 1
    blist, c = (1,), 1
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c = c*(4*i+2)//(i+2)
A000736_list = list(islice(A000736_gen(),40)) # Chai Wah Wu, Jun 12 2022

E.g.f.: (sec(x) + tan(x))*(integral(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)),x)+1). - Sergei N. Gladkovskii, Oct 30 2014

a(n) ~ n! * (6/Pi+2*exp(Pi)*((2-1/Pi)*BesselI(0,Pi)-2*BesselI(1,Pi))) * 2^n / Pi^n. - Vaclav Kotesovec, Oct 30 2014

A000746 Boustrophedon transform of triangular numbers.

Original entry on oeis.org

1, 4, 13, 39, 120, 407, 1578, 7042, 35840, 205253, 1306454, 9148392, 69887664, 578392583, 5155022894, 49226836114, 501420422112, 5426640606697, 62184720675718, 752172431553308, 9576956842743904, 128034481788227195

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. A000217, A000718, A109449.

a000746 n = sum $ zipWith (*) (a109449_row n) $ tail a000217_list
-- Reinhard Zumkeller, Nov 03 2013

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( (Sec(x)+Tan(x))*Exp(x)*(x^2+4*x+2)/2 ))); // G. C. Greubel, Jul 10 2025

t[n_, 0] := (n + 1) (n + 2)/2; 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 *)

from itertools import accumulate, count, islice
def A000746_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(2):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c += i
A000746_list = list(islice(A000746_gen(),40)) # Chai Wah Wu, Jun 12 2022

@CachedFunction
def f(n, k):
    if (k==0): return binomial(n+2,2)
    else: return f(n, k-1) + f(n-1, n-k)
def A000746(n): return f(n,n)
[A000746(n) for n in range(31)] # G. C. Greubel, Jul 10 2025

a(n) = Sum_{k=0..n} A109449(n,k)*(k + 1)*(k + 2)/2. - Reinhard Zumkeller, Nov 03 2013
E.g.f.: (sec(x) + tan(x))*exp(x)*(x^2 + 4*x + 2)/2. - Sergei N. Gladkovskii, Oct 30 2014
a(n) ~ n! * (Pi^2 + 8*Pi + 8) * exp(Pi/2) * 2^(n-1) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

A230952 Boustrophedon transform of Hamming weight (A000120).

Original entry on oeis.org

0, 1, 3, 8, 23, 72, 280, 1242, 6331, 36236, 230726, 1615584, 12342422, 102145644, 910393530, 8693609421, 88552405435, 958361506524, 10982014291650, 132835979792636, 1691320230842116, 22611285878526978, 316685416851528722, 4636988553066906265

Offset: 0

Reinhard Zumkeller, Nov 03 2013

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

Cf. A230950, A230951.

a230952 n = sum $ zipWith (*) (a109449_row n) $ map fromIntegral a000120_list
(Python 3.10+)
from itertools import accumulate, count, islice
def A230952_gen(): # generator of terms
        blist = tuple()
        for i in count(0):
            yield (blist := tuple(accumulate(reversed(blist),initial=i.bit_count())))[-1]
A230952_list = list(islice(A230952_gen(),40)) # Chai Wah Wu, Jun 12 2022

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] DigitCount[k, 2, 1], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 23 2019 *)

a(n) = Sum_{k=0..n} A109449(n,k)*A000120(k).

A230957 Boustrophedon transform of partition numbers A000009.

Original entry on oeis.org

1, 2, 4, 10, 29, 94, 364, 1621, 8255, 47277, 300962, 2107479, 16099922, 133243363, 1187555333, 11340314638, 115511502857, 1250127378307, 14325404633040, 173276880401035, 2206229765086251, 29495119298584886, 413097874985119467, 6048684327982905454

Offset: 0

Reinhard Zumkeller, Nov 03 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..150
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. A000751.

a230957 n = sum $ zipWith (*) (a109449_row n) a000009_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] PartitionsQ[k], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 23 2019 *)

a(n) = sum(A109449(n,k)*A000009(k): k=0..n).

A231200 Boustrophedon transform of even numbers.

Original entry on oeis.org

0, 2, 8, 24, 72, 240, 924, 4116, 20944, 119952, 763540, 5346748, 40845816, 338041704, 3012855356, 28770647220, 293055401888, 3171602665696, 36343889387172, 439607533130732, 5597256953340360, 74829813397495128, 1048039052970587788, 15345654816688856484

Offset: 0

Reinhard Zumkeller, Nov 05 2013

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

Cf. A005843, A000754.

a231200 n = sum $ zipWith (*) (a109449_row n) $ [0, 2 ..]

T[n_, k_] := SeriesCoefficient[(1+Sin[x])/Cos[x], {x, 0, n-k}] n!/k!;
a[n_] := 2 Sum[k T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 28 2019 *)

from itertools import accumulate, count, islice
def A231200_gen(): # generator of terms
    blist = tuple()
    for i in count(0,2):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A231200_list = list(islice(A231200_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = Sum_{k=0..n} A109449(n,k)*k*2.
a(n) = 2*A231179(n).
E.g.f.: 2*x*exp(x)*(sec(x) + tan(x)). - Ilya Gutkovskiy, Sep 27 2017

A000660 Boustrophedon transform of 1,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 5, 14, 41, 136, 523, 2330, 11857, 67912, 432291, 3027166, 23125673, 191389108, 1705788659, 16289080922, 165919213089, 1795666675824, 20576824369027, 248892651678198, 3168999664907705, 42366404751871660, 593368400878431795, 8688251294851280594

Offset: 0

N. J. A. Sloane, Simon Plouffe

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

Cf. A000111, A028310, A109449, A231179.

a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])
-- Reinhard Zumkeller, Nov 04 2013

seq(coeff(series(factorial(n)*(x*exp(x)+1)*(sec(x)+tan(x)), x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 30 2018

a[n_] := n! SeriesCoefficient[(1+x Exp[x])(1+Sin[x])/Cos[x], {x, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 30 2018, after Sergei N. Gladkovskii *)

from itertools import accumulate, count, islice
def A000660_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000660_list = list(islice(A000660_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Algorithm of L. Seidel (1877)
def A000660_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in range(n) :
        Am = i
        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
A000660_list(10) # Peter Luschny, Jun 02 2012

a(n) = Sum_{k=0..n} A109449(n,k)*A028310(k). - Reinhard Zumkeller, Nov 04 2013
E.g.f.: (x*exp(x) + 1)*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Oct 28 2014
a(n) = A231179(n) + A000111(n). - Sergei N. Gladkovskii, Oct 28 2014
a(n) ~ n! * (2 + Pi*exp(Pi/2)) * (2/Pi)^(n+1). - Vaclav Kotesovec, Jun 12 2015

A000733 Boustrophedon transform of partition numbers 1, 1, 1, 2, 3, 5, 7, ...

Original entry on oeis.org

1, 2, 4, 10, 30, 101, 394, 1760, 8970, 51368, 326991, 2289669, 17491625, 144760655, 1290204758, 12320541392, 125496010615, 1358185050788, 15563654383395, 188254471337718, 2396930376564860, 32044598671291610

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

			The array begins:
                   1
               1  ->   2
           4  <-   3  <-   1
       2  ->   6  ->   9  ->  10
  30  <-  28  <-  22  <-  13  <-   3
- _John Cerkan_, Jan 26 2017

John Cerkan, Table of n, a(n) for n = 0..482
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. A000041, A000751, A109449, A230957.

a000733 n = sum $ zipWith (*) (a109449_row n) (1 : a000041_list)
-- Reinhard Zumkeller, Nov 04 2013

t[n_, 0] := If[n == 0, 1, PartitionsP[n-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 *)

from itertools import count, accumulate, islice
from sympy import npartitions
def A000733_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=npartitions(i))))[-1]
A000733_list = list(islice(A000733_gen(),40)) # Chai Wah Wu, Jun 12 2022

A230958 Boustrophedon transform of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 3, 7, 15, 39, 127, 480, 2143, 10907, 62495, 397814, 2785861, 21282228, 176133285, 1569817724, 14990658724, 152693582275, 1652531857935, 18936620009722, 229053108410969, 2916394751599614, 38989325834726043, 546070266163669664, 7995699956778626764

Offset: 0

Reinhard Zumkeller, Nov 04 2013

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 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]
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A029885, A230950, A230951.

Cf. A001285, A109449.

a230958 n = sum $ zipWith (*) (a109449_row n) $ map fromIntegral a001285_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
tm[n_] := Mod[Sum[Mod[Binomial[n, k], 2], {k, 0, n}], 3];
Table[Sum[T[n, k] tm[k], {k, 0, n}], {n, 0, 23}] (* Jean-François Alcover, Jul 23 2019 *)

from itertools import accumulate, count, islice
def A230958_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist), initial=2 if i.bit_count()&1 else 1)))[-1]
A230958_list = list(islice(A230958_gen(),30)) # Chai Wah Wu, Apr 17 2023

a(n) = Sum_{k=0..n} A109449(n,k)*A001285(k).

A000674 Boustrophedon transform of 1, 2, 2, 2, 2, ...

Original entry on oeis.org

1, 3, 7, 16, 43, 138, 527, 2346, 11943, 68418, 435547, 3050026, 23300443, 192835698, 1718682167, 16412205306, 167173350543, 1809239622978, 20732358910387, 250773962554186, 3192953259262243, 42686640718266258, 597853508941160207

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

			G.f. = 1 + 3*x + 7*x^2 + 16*x^3 + 43*x^4 + 138*x^5 + 527*x^6 + 2346*x^7 + ...

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. A000111, A040000, A109449.

a000674 n = sum $ zipWith (*) (a109449_row n) (1 : repeat 2)
-- Reinhard Zumkeller, Nov 04 2013

With[{nn=30},CoefficientList[Series[(Sec[x]+Tan[x])(2Exp[x]-1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2015 *)

from itertools import accumulate, islice
def A000674_gen(): # generator of terms
    yield 1
    blist = (1,)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=2)))[-1]
A000674_list = list(islice(A000674_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A040000(k). - Reinhard Zumkeller, Nov 04 2013
E.g.f.: (sec(x) + tan(x))*(2*exp(x) - 1). - Sergei N. Gladkovskii, Oct 28 2014
Binomial convolution of A000111 and A040000. - Michael Somos, Oct 30 2014
a(n) ~ n! * (2*exp(Pi/2)-1) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

More terms from Sean A. Irvine, Feb 20 2011

A173253 Partial sums of A000111.

Original entry on oeis.org

1, 2, 3, 5, 10, 26, 87, 359, 1744, 9680, 60201, 413993, 3116758, 25485014, 224845995, 2128603307, 21520115452, 231385458428, 2636265133869, 31725150246701, 402096338484226, 5353594391608322, 74702468784746223, 1090126355291598575, 16604660518848685480

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

Partial sums of Euler or up/down numbers. Partial sums of expansion of sec x + tan x. Partial sums of number of alternating permutations on n letters.

			a(22) = 1 + 1 + 1 + 2 + 5 + 16 + 61 + 272 + 1385 + 7936 + 50521 + 353792 + 2702765 + 22368256 + 199360981 + 1903757312 + 19391512145 + 209865342976 + 2404879675441 + 29088885112832 + 370371188237525 + 4951498053124096 + 69348874393137901.

Alois P. Heinz, Table of n, a(n) for n = 0..485

Cf. A000111, A000364, A000182, A008280, A008281, A008282, A010094, A059720, A008970, A109449, A162170.

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) option remember;
      `if`(n<0, 0, a(n-1))+ b(n, 0)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 27 2017

With[{nn=30},Accumulate[CoefficientList[Series[Sec[x]+Tan[x],{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Feb 26 2012 *)

from itertools import accumulate
def A173253(n):
    if n<=1:
        return n+1
    c, blist = 2, (0,1)
    for _ in range(n-1):
        c += (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
    return c # Chai Wah Wu, Apr 16 2023

a(n) = SUM[i=0..n] A000111(i) = SUM[i=0..n] (2^i|E(i,1/2)+E(i,1)| where E(n,x) are the Euler polynomials).
G.f.: (1 + x/Q(0))/(1-x),m=+4,u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
G.f.: 1/(1-x) + T(0)*x/(1-x)^2, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2013
a(n) ~ 2^(n+2)*n!/Pi^(n+1). - Vaclav Kotesovec, Oct 27 2016

cp's OEIS Frontend

A000736 Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...

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A000746 Boustrophedon transform of triangular numbers.

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A230952 Boustrophedon transform of Hamming weight (A000120).

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A230957 Boustrophedon transform of partition numbers A000009.

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A231200 Boustrophedon transform of even numbers.

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A000660 Boustrophedon transform of 1,1,2,3,4,5,...

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A000733 Boustrophedon transform of partition numbers 1, 1, 1, 2, 3, 5, 7, ...

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