A098830 - OEIS

%I #69 Oct 19 2023 19:45:39

%S 0,1,2,4,10,32,126,588,3170,19384,132550,1002212,8301930,74767056,

%T 727348814,7601002876,84920459890,1010058659048,12742908917718,

%U 169962226236180,2389587638934650,35321010036943360,547577222471444062

%N 3*Sum_{k>=0} k^n/binomial(2*k, k) = Pi*sqrt(3)*q(n) + a(n) for some rational sequence (q(n)).

%C For n >= 0, this appears to be the number of permutations on n+1 elements having the "ascending-to-max" property (see He et al., Definition 2.1). - _Nathaniel Johnston_, Apr 10 2011

%C a(n) is the number of permutations of [n] in which the excedance entries are precisely the entries to the left of 1. For example a(3) = 4 counts 123, 213, 231, 312, but does not count 132 because 3 is an excedance not to the left of 1, or 321 because 2 is not an excedance. See link at A099594. - _David Callan_, Dec 14 2021

%H Alois P. Heinz, <a href="/A098830/b098830.txt">Table of n, a(n) for n = -1..180</a>

%H Beata Bényi and Peter Hajnal, <a href="https://arxiv.org/abs/1602.08684">Combinatorial properties of poly-Bernoulli relatives</a>, arXiv preprint arXiv:1602.08684 [math.CO], 2016.

%H Beáta Bényi and Toshiki Matsusaka, <a href="https://arxiv.org/abs/2106.05585">Extensions of the combinatorics of poly-Bernoulli numbers</a>, arXiv:2106.05585 [math.CO], 2021.

%H Beáta Bényi and Toshiki Matsusaka, <a href="https://doi.org/10.2206/kyushujm.77.149">Remarkable relations between the central binomial series, Eulerian polynomials, and poly-Bernoulli numbers</a>, Kyushu J. Math. 77 (2023), 149-158. <a href="https://arxiv.org/abs/2207.00205">On arXiv</a>, arXiv:2207.00205 [math.NT], 2022.

%H Meng He, J. Ian Munro, and S. Srinivasa Rao, <a href="http://web.cs.dal.ca/~mhe/publications/soda05_suffixarray.pdf">A Categorization Theorem on Suffix Arrays with Applications to Space Efficient Text Indexes</a>, SODA 2005.

%F a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (j+1)^k*Sum_{i=0..j} (-1)^(n-k+j-i)*C(j, i)*(j-i)^(n-k). - _Paul D. Hanna_, Nov 03 2004

%F a(n) ~ Pi * n^(n+1) / (exp(n) * 2^n * (log(2))^(n+3/2)). - _Vaclav Kotesovec_, Sep 08 2014

%F E.g.f: 12*exp(x/2)*(arcsin(exp(x/2)/2)-Pi/6)/(4-exp(x))^(3/2) + 12/(4-exp(x)) - 3. - _Robert Israel_, Nov 27 2014

%F a(n) = Sum_{k=0..n} Sum_{m=0..k} (-1)^(k+m)*(m+1)^(n-k)*m!*Stirling2(k,m). - _Vladimir Reshetnikov_, Dec 17 2015

%e 3*Sum_{k>=0} k^3/binomial(2k, k) = (238/81)*Pi*sqrt(3) + 32, hence a(4)=32.

%p egf:= 12*exp(x/2)*(arcsin(exp(x/2)/2)-Pi/6)/(4-exp(x))^(3/2) + 12/(4-exp(x)) - 3:

%p S:= series(egf,x,101):

%p 0,seq(coeff(S,x,j)*j!,j=0..100); # _Robert Israel_, Nov 27 2014

%p a := proc(n) option remember; if n < 0 then 0 else

%p 1 + (2*a(n-1) + add(binomial(n, k)*a(k), k = 0..n-1))/3 fi end:

%p seq(a(n), n = -1..21); # _Peter Luschny_, Aug 01 2021

%t a[n_] := Sum[(j+1)^k*Sum[(-1)^(n-k+j-i)*If[i == j && n == k, 1, (j-i)^(n-k)]*Binomial[j, i], {i, 0, j}], {k, 0, n}, {j, 0, n-k}]; Table[a[n], {n, -1, 21}] (* _Jean-François Alcover_, Jan 22 2014, after _Paul D. Hanna_ *)

%t Table[Sum[(-1)^(k+m) (m+1)^(n-k) m! StirlingS2[k, m], {k, 0, n}, {m, 0, k}], {n, -1, 20}] (* _Vladimir Reshetnikov_, Dec 17 2015 *)

%o (PARI) {a(n)=sum(k=0,n,sum(j=0,n-k,(j+1)^k*sum(i=0,j,(-1)^(n-k+j-i)*binomial(j,i)*(j-i)^(n-k))))}

%Y See also A181334 and A185585.

%Y Antidiagonal sums of array in A099594. - _Ralf Stephan_, Oct 28 2004

%K nonn

%O -1,3

%A _Benoit Cloitre_, Oct 09 2004