A109449 -id:A109449 - cp's OEIS frontend

A000753 Boustrophedon transform of Catalan numbers.

1, 2, 5, 16, 59, 243, 1101, 5461, 29619, 175641, 1137741, 8031838, 61569345, 510230087, 4549650423, 43452408496, 442620720531, 4790322653809, 54893121512453, 663974736739232, 8453986695437957, 113021461431438475

Offset: 0

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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

Cf. A000108, A000736, A109449.

a000753 n = sum $ zipWith (*) (a109449_row n) a000108_list
-- Reinhard Zumkeller, Nov 05 2013

CoefficientList[Series[E^(2*x) * (BesselI[0,2*x] - BesselI[1,2*x]) * (Sec[x] + Tan[x]),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Oct 30 2014 after Sergei N. Gladkovskii *)

from itertools import accumulate, count, islice
def A000753_gen(): # generator of terms
    blist, c = tuple(), 1
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
        c = c*(4*i+2)//(i+2)
A000753_list = list(islice(A000753_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A000108(k). - Reinhard Zumkeller, Nov 05 2013
E.g.f.: (sec(x) + tan(x))*exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - Sergei N. Gladkovskii, Oct 30 2014
a(n) ~ n! * exp(Pi) * (BesselI(0, Pi) - BesselI(1, Pi)) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Oct 30 2014

A231179 Boustrophedon transform of nonnegative integers, cf. A001477.

Original entry on oeis.org

0, 1, 4, 12, 36, 120, 462, 2058, 10472, 59976, 381770, 2673374, 20422908, 169020852, 1506427678, 14385323610, 146527700944, 1585801332848, 18171944693586, 219803766565366, 2798628476670180, 37414906698747564, 524019526485293894, 7672827408344428242

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. A000111, A000737, A000660, A109449, A231200.

a231179 n = sum $ zipWith (*) (a109449_row n) [0..]

a[n_] := n! SeriesCoefficient[x Exp[x] (1+Sin[x]) / Cos[x], {x, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *)

from itertools import count, islice, accumulate
def A231179_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A231179_list = list(islice(A231179_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = A231200(n)/2.
a(n) = Sum_{k=1..n} k * A109449(n,k).
E.g.f.: x*exp(x)*(sec(x)+tan(x)). (After Sergei N. Gladkovskii in A000660.) - Peter Luschny, Oct 28 2014
a(n) = A000660(n) - A000111(n). - Sergei N. Gladkovskii, Oct 28 2014
a(n) ~ n! * exp(Pi/2) * 2^(n+1) / Pi^n. - 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

A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, -2, 0, 3, 0, 0, -8, 0, 4, 0, 16, 0, -20, 0, 5, 0, 0, 96, 0, -40, 0, 6, 0, -272, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0, 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0, 0, 79360, 0, -32640, 0, 4032, 0, -240, 0, 10, 0

Offset: 0

Peter Luschny, Jul 09 2009

Definition. V_n(x) = (skp(n, x+1) - skp(n, x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
Equivalently, let the polynomials V_n(x) (n>=0) defined by V_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*L(k)*(x+v+1)^n; the sequence L(k) = -1 - H(k-1)*(-1)^floor((k-1)/4) / 2^floor(k/2) if k > 0 and L(0)=0; H(k) = 1 if k mod 4 <> 0, otherwise 0.
(1) V_n(0) = 2^n * Euler(n,1) for n > 0, A155585.
(2) V_n(1) = 1 - Euler(n).
(3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642.
(4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1.
(5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586.
The Swiss-Knife polynomials (A153641) are complementary to the polynomials defined here. Adding both gives polynomials with e.g.f. exp(x*t)*(sech(t)+tanh(t)), the coefficients of which are a signed variant of A109449.
The Swiss-Knife polynomials as well as the complementary Swiss-Knife polynomials are closely related to the Bernoulli and Euler polynomials. Let F be a sequence and
P_{F}[n](x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n.
V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials,
W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials.
B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials,
E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials.
The most striking formal difference between the Swiss-Knife-type polynomials and the Bernoulli-Euler type polynomials is: The SK-type polynomials have integer coefficients whereas the BE-type polynomials have rational coefficients.
Let R be the exponential Riordan array (exp(x)*sech(x), x) = P * A119879 = 2*P(I + P^2)^(-1) where P denotes Pascal's triangle A007318. Then T = R - I. - Peter Bala, Mar 07 2024

			Triangle begins:
  [0]    0;
  [1]    1,     0;
  [2]    0,     2,     0;
  [3]   -2,     0,     3,   0;
  [4]    0,    -8,     0,   4,    0;
  [5]   16,     0,   -20,   0,    5,    0;
  [6]    0,    96,     0, -40,    0,    6,    0;
  [7] -272,     0,   336,   0,  -70,    0,    7,  0;
  [8]    0, -2176,     0, 896,    0, -112,    0,  8,  0;
  [9] 7936,     0, -9792,   0, 2016,    0, -168,  0,  9,  0;

Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife Polynomials and Euler Numbers.
Wikipedia, Bernoulli number.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

V_n(k), n=0, 1, ..., k=0: A155585, k=1: A009832,
V_n(k), k=0, 1, ..., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536.
Cf. A153641, A154341, A154342, A154343, A154344, A154345.

# Polynomials V_n(x):
V := proc(n,x) local k,pow; pow := (n,k) -> `if`(n=0 and k=0,1,n^k); add(binomial(n,k)*euler(k)*pow(x+1,n-k),k=0..n) - pow(x,n) end:
# Coefficients a(n):
seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t),t,16),t,n),x,k),k=0..n)),n=0..8);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; v[n_, x_] := (skp[n, x+1]-skp[n, x-1])/2; t[n_, k_] := Coefficient[v[n, x], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

R = PolynomialRing(QQ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
    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 A162660(n,k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k]
matrix(ZZ, 9, A162660) # Peter Luschny, Jul 23 2012

T(n, k) = [x^(n-k)](skp(n,x+1)-skp(n,x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
E.g.f. exp(x*t)*tanh(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2-2)*(t^3/3!) + ...
V_n(x) = -x^n + Sum_{k=0..n} C(n,k)*Euler(k)*(x+1)^(n-k).

A230953 Boustrophedon transform of odd primes, cf. A065091.

Original entry on oeis.org

3, 8, 20, 53, 154, 505, 1944, 8651, 44046, 252271, 1605874, 11245261, 85907084, 710970323, 6336648426, 60510526207, 616355168958, 6670526004559, 76438597647616, 924584128977111, 11772170758462928, 157382330019694067, 2204239468545788024, 32275035859881159165

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. A000747, A230956, A230954, A230955.

a230953 n = sum $ zipWith (*) (a109449_row n) $ tail a000040_list

t[n_, 0] := Prime[n+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
from sympy import prime
def A230953_gen(): # generator of terms
    blist = tuple()
    for i in count(2):
        yield (blist := tuple(accumulate(reversed(blist),initial=prime(i))))[-1]
A230953_list = list(islice(A230953_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A000040(k+2).

E.g.f.: (sec(x) + tan(x)) * Sum_{k>=0} prime(k+2)*x^k/k!. - Ilya Gutkovskiy, Jun 26 2018

A230954 Boustrophedon transform of composite numbers.

Original entry on oeis.org

4, 10, 24, 59, 162, 526, 2016, 8978, 45696, 261760, 1666308, 11668652, 89141580, 737740174, 6575238599, 62788901290, 639562492327, 6921688171153, 79316703841369, 959397056891093, 12215422719238138, 163308172248420391, 2287234651735371577

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. A230955, A000747, A000732, A230953.

a230954 n = sum $ zipWith (*) (a109449_row n) a002808_list

cc = Select[Range[max = 40], CompositeQ]; t[n_, 0] := cc[[n+1]]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, cc // Length, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
from sympy import composite
def A230954_gen(): # generator of terms
    blist = tuple()
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=composite(i))))[-1]
A230954_list = list(islice(A230954_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = sum(A109449(n,k)*A002808(k+1): k=0..n).

A230955 Boustrophedon transform of nonprimes.

Original entry on oeis.org

1, 5, 15, 40, 114, 371, 1422, 6334, 32238, 184655, 1175454, 8231308, 62882262, 520416569, 4638303786, 44292536061, 451160065069, 4882696090609, 55951575728713, 676777708544967, 8617001415386120, 115200823068725262, 1613460678695102980, 23624702309844184487

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. A230954, A000747, A000732, A230953.

a230955 n = sum $ zipWith (*) (a109449_row n) a018252_list

cc = Select[Range[max = 40], !PrimeQ[#]&]; t[n_, 0] := cc[[n+1]]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, cc // Length, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, count, islice
from sympy import composite
def A230955_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=composite(i))))[-1]
A230955_list = list(islice(A230955_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A018252(k+1).

A000744 Boustrophedon transform (second version) of Fibonacci numbers 1,1,2,3,...

Original entry on oeis.org

1, 2, 5, 14, 42, 144, 563, 2526, 12877, 73778, 469616, 3288428, 25121097, 207902202, 1852961189, 17694468210, 180234349762, 1950592724756, 22352145975707, 270366543452702, 3442413745494957, 46021681757269830

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 144*x^5 + 563*x^6 + 2526*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 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, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A000045, A000687, A000738, A092073.

a000744 n = sum $ zipWith (*) (a109449_row n) $ tail a000045_list
-- Reinhard Zumkeller, Nov 03 2013

read(transforms);
with(combinat):
F:=fibonacci;
[seq(F(n), n=1..50)];
BOUS2(%);

s[k_] := SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, k}] k!;
b[n_, k_] := Binomial[n, k] s[n - k];
a[n_] := Sum[b[n, k] Fibonacci[k + 1], {k, 0, n}];
Array[a, 22, 0] (* Jean-François Alcover, Jun 01 2019 *)

from itertools import accumulate, islice
def A000744_gen(): # generator of terms
    blist, a, b = tuple(), 1, 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
        a, b = b, a+b
A000744_list = list(islice(A000744_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A000045(k+1). - Reinhard Zumkeller, Nov 03 2013
E.g.f.: (1/10)*(sec(x)+tan(x))*((5^(1/2)+1)*exp(1/2*x*(5^(1/2)+1))+(5^(1/2)-1)*exp(1/2*x*(-5^(1/2)+1)))*5^(1/2). - Sergei N. Gladkovskii, Oct 30 2014
a(n) ~ n! * (sqrt(5) - 1 + (1+sqrt(5)) * exp(sqrt(5)*Pi/2)) * 2^(n+1) / (sqrt(5) * exp((sqrt(5)-1)*Pi/4) * Pi^(n+1)). - Vaclav Kotesovec, Jun 12 2015

Entry revised by N. J. A. Sloane, Mar 16 2011

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]

A230961 Boustrophedon transform of factorials beginning with 1, cf. A000142.

Original entry on oeis.org

1, 3, 11, 50, 273, 1746, 12823, 106462, 986689, 10103074, 113309991, 1381835454, 18209834849, 257911743506, 3907538236631, 63066584719982, 1080340925760129, 19577690297352258, 374214932301757255, 7524626434657416286, 158783753482817132065

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. A230960.

a230961 n = sum $ zipWith (*) (a109449_row n) $ tail a000142_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] (k + 1)!, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 23 2019 *)

from itertools import accumulate, count, islice
def A230961_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := m*i))))[-1]
A230961_list = list(islice(A230961_gen(),40)) # Chai Wah Wu, Jun 12 2022

a(n) = sum(A109449(n,k)*A000142(k+1): k=0..n).
E.g.f.: conjecture: (tan(x)+sec(x))/(1-2*x+x^2) = (1- 12*x/ (Q(0)+6*x+3*x^2))/(1-x)^2, where Q(k) = 2*(4*k+1)*(32*k^2+16*k - x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) ~ n! * n * (1+sin(1))/cos(1). - Vaclav Kotesovec, Jun 12 2015

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