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A099960 An interleaving of the Genocchi numbers of the first and second kind, A110501 and A005439. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A148036 A148037 A290383 * A324963 A218090 A101182

Adjacent sequences:  A099957 A099958 A099959 * A099961 A099962 A099963

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 13 2004, following a suggestion made by Douglas G. Rogers, Mar 10 2003

EXTENSIONS

More terms from Emeric Deutsch, Nov 16 2004

STATUS

approved

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Last modified May 25 23:54 EDT 2019. Contains 323576 sequences. (Running on oeis4.)