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A217924 Row sums of triangle A217537. 1
1, 1, 3, 9, 35, 153, 755, 4105, 24323, 155513, 1064851, 7760745, 59895203, 487397849, 4166564147, 37298443977, 348667014723, 3395240969785, 34365336725715, 360837080222761, 3923531021460707, 44108832866004121, 511948390801374835, 6126363766802713481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The inverse binomial transform of a(n) is A194689.

A087981(n)  =  Sum((-1)^k*s(n+1,k+1)*a(k) for k in (0..n))

|A000023(n)| = |Sum((-1)^(n-k)*s(n,k)*a(k) for k in (0..n))|

where s(n,k) are the unsigned Stirling numbers of first kind.

a(n) is the number of inequivalent set partitions of {1,2,...,n} where two blocks are considered equivalent when one can be obtained from the other by an alternating (even) permutation. - Geoffrey Critzer, Mar 17 2013

LINKS

Table of n, a(n) for n=0..23.

FORMULA

G.f.: 1/Q(0) where Q(k) =  1 + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013

E.g.f.: exp(2*exp(x) - x - 2). -  Geoffrey Critzer, Mar 17 2013

G.f.: 1/Q(0), where Q(k)= 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013

G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-x*k)*(1-2*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013

a(n) = sum(k=0..n, sum(j=0..k, binomial(n,k-j)*2^j*(-1)^(k-j)*stirling2(n-k+j,j))). - Vladimir Kruchinin, Feb 28 2015

EXAMPLE

a(3)=9 because we have: {1,2,3}; {1,3,2}; {1}{2,3}; {1}{3,2}; {2}{1,3}; {2}{3,1}; {3}{1,2}; {3}{2,1}; {1}{2}{3}. [Geoffrey Critzer, Mar 17 2013]

MATHEMATICA

nn=23; Range[0, nn]!CoefficientList[Series[Exp[2 Exp[x]-x-2], {x, 0, nn}], x]  (* Geoffrey Critzer, Mar 17 2013 *)

nmax = 25; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-2*k*x^2 , 1 - (k + 1)*x, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 25 2017 *)

PROG

(Sage)

def A217924_list(n):

    T = A217537_triangle(n)

    return [add(T.row(n)) for n in range(n)]

A217924_list(24)

(Maxima)

a(n):=sum(sum(binomial(n, k-j)*2^j*(-1)^(k-j)*stirling2(n-k+j, j), j, 0, k), k, 0, n); /* Vladimir Kruchinin, Feb 28 2015 */

CROSSREFS

Sequence in context: A151045 A225041 A074507 * A030268 A097277 A034428

Adjacent sequences:  A217921 A217922 A217923 * A217925 A217926 A217927

KEYWORD

nonn

AUTHOR

Peter Luschny, Oct 15 2012

STATUS

approved

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Last modified August 18 13:12 EDT 2019. Contains 326100 sequences. (Running on oeis4.)