A098830 3*Sum_{k>=0} k^n/binomial(2*k, k) = Pi*sqrt(3)*q(n) + a(n) for some rational sequence (q(n)).

0, 1, 2, 4, 10, 32, 126, 588, 3170, 19384, 132550, 1002212, 8301930, 74767056, 727348814, 7601002876, 84920459890, 1010058659048, 12742908917718, 169962226236180, 2389587638934650, 35321010036943360, 547577222471444062

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

See also A181334 and A185585.

Antidiagonal sums of array in A099594. - Ralf Stephan, Oct 28 2004

Sequence in context: A263665 A001250 A013032 * A121277 A009284 A105557

Adjacent sequences: A098827 A098828 A098829 * A098831 A098832 A098833

Keyword

nonn

Author

Benoit Cloitre, Oct 09 2004

Status

approved