A099960 An interleaving of the Genocchi numbers of the first and second kind, A110501 and A005439.

1, 1, 1, 1, 2, 3, 8, 17, 56, 155, 608, 2073, 9440, 38227, 198272, 929569, 5410688, 28820619, 186043904, 1109652905, 7867739648, 51943281731, 401293838336, 2905151042481, 24290513745920, 191329672483963, 1721379917619200, 14655626154768697, 141174819474169856

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

Cf. A022493, A099959, A001469, A005439, A138265, A294281.

Sequence in context: A148037 A345324 A290383 * A324963 A218090 A101182

Adjacent sequences: A099957 A099958 A099959 * A099961 A099962 A099963

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 13 2004

Extensions

More terms from Emeric Deutsch, Nov 16 2004

Status

approved