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A062162
Boustrophedon transform of (-1)^n.
9
1, 0, 0, 1, 0, 5, 10, 61, 280, 1665, 10470, 73621, 561660, 4650425, 41441530, 395757181, 4031082640, 43626778785, 499925138190, 6046986040741, 76992601769220, 1029315335116745, 14416214547400450, 211085887742964301, 3225154787165157400, 51329932704636904305
OFFSET
0,6
COMMENTS
Inverse binomial transform of Euler numbers A000111. - Paul Barry, Jan 21 2005
a(n) = abs(sum of row n in A247453). - Reinhard Zumkeller, Sep 17 2014
LINKS
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]
FORMULA
E.g.f.: exp(-x)*(tan(x) + sec(x)). - Vladeta Jovovic, Feb 11 2003
a(n) ~ 4*(2*n/Pi)^(n+1/2)/exp(n+Pi/2). - Vaclav Kotesovec, Oct 05 2013
G.f.: E(0)*x/(1+x) + 1/(1+x), where E(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(x*k-1)*(x*(k+1)-1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014
MATHEMATICA
CoefficientList[Series[E^(-x)*(Tan[x]+1/Cos[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)
t[n_, 0] := (-1)^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 *)
PROG
(Sage) # Generalized algorithm of L. Seidel (1877)
def A062162_list(n) :
R = []; A = {-1:0, 0:0}
k = 0; e = 1
for i in range(n) :
Am = (-1)^i
A[k + e] = 0
e = -e
for j in (0..i) :
Am += A[k]
A[k] = Am
k += e
R.append(A[e*i//2])
return R
A062162_list(22) # Peter Luschny, Jun 02 2012
(Haskell)
a062162 = abs . sum . a247453_row -- Reinhard Zumkeller, Sep 17 2014
(Python)
from itertools import islice, accumulate
def A062162_gen(): # generator of terms
blist, m = tuple(), -1
while True:
yield (blist := tuple(accumulate(reversed(blist), initial=(m:=-m))))[-1]
A062162_list = list(islice(A062162_gen(), 20)) # Chai Wah Wu, Jun 10 2022
CROSSREFS
Cf. A000111 (binomial transform).
Cf. A000667.
Cf. A247453.
Sequence in context: A072309 A328332 A240647 * A062848 A054884 A061518
KEYWORD
nonn,easy
AUTHOR
Frank Ellermann, Jun 10 2001
STATUS
approved