OFFSET
0,5
COMMENTS
First column (also row sums) of triangle in A099959.
Number of ascent sequences of length n without level steps and with alternating ascents and descents. a(6) = 8: 010101, 010102, 010103, 010201, 010202, 010203, 010212, 010213. - Alois P. Heinz, Oct 27 2017
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, p. 220, answer to exercise 174, Addison-Wesley, 2009.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
Catalin Zara, Cardinality of l_1-Segments and Genocchi Numbers, arXiv:1304.5798 [math.CO] (2013)
FORMULA
a(n) ~ 2^(5/2) * n^(n+3/2) / (Pi^(n+1/2) * exp(n)). - Vaclav Kotesovec, Sep 10 2014
MAPLE
with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n, p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j], j=1..i): vector(n, q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j], j=1..i) else sum(a[j], j=1..n) fi end: vector(n+1, q) end: R[0]:=vector(1, 1): for n from 1 to 30 do if n mod 2 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: seq(R[n][1], n=0..30); # Emeric Deutsch
MATHEMATICA
g1 = Table[2*(4^n-1)*BernoulliB[2*n] // Abs, {n, 0, 13}]; g2 = Table[2*(-1)^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}], {n, 0, 13}]; Riffle[g1, g2] // Rest (* Jean-François Alcover, May 23 2013 *)
PROG
(Sage) # Algorithm of L. Seidel (1877)
def A099960_list(n) :
D = [0]*(n//2+3); D[1] = 1
R = []; b = True; h = 1
for i in (1..n) :
if b :
for k in range(h, 0, -1) : D[k] += D[k+1]
R.append(D[1]); h += 1
else :
for k in range(1, h, 1) : D[k] += D[k-1]
R.append(D[h-1])
b = not b
return R
A099960_list(27) # Peter Luschny, Apr 30 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 13 2004
EXTENSIONS
More terms from Emeric Deutsch, Nov 16 2004
STATUS
approved