OFFSET
-1,3
COMMENTS
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
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = -1..180
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016.
Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021.
Beáta Bényi and Toshiki Matsusaka, Remarkable relations between the central binomial series, Eulerian polynomials, and poly-Bernoulli numbers, Kyushu J. Math. 77 (2023), 149-158. On arXiv, arXiv:2207.00205 [math.NT], 2022.
Meng He, J. Ian Munro, and S. Srinivasa Rao, A Categorization Theorem on Suffix Arrays with Applications to Space Efficient Text Indexes, SODA 2005.
FORMULA
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
a(n) ~ Pi * n^(n+1) / (exp(n) * 2^n * (log(2))^(n+3/2)). - Vaclav Kotesovec, Sep 08 2014
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
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
EXAMPLE
3*Sum_{k>=0} k^3/binomial(2k, k) = (238/81)*Pi*sqrt(3) + 32, hence a(4)=32.
MAPLE
egf:= 12*exp(x/2)*(arcsin(exp(x/2)/2)-Pi/6)/(4-exp(x))^(3/2) + 12/(4-exp(x)) - 3:
S:= series(egf, x, 101):
0, seq(coeff(S, x, j)*j!, j=0..100); # Robert Israel, Nov 27 2014
a := proc(n) option remember; if n < 0 then 0 else
1 + (2*a(n-1) + add(binomial(n, k)*a(k), k = 0..n-1))/3 fi end:
seq(a(n), n = -1..21); # Peter Luschny, Aug 01 2021
MATHEMATICA
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 *)
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 *)
PROG
(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))))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Oct 09 2004
STATUS
approved