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 A098830 3*Sum_{k>=0} k^n/binomial(2*k, k) = Pi*sqrt(3)*q(n) + a(n) for some rational sequence (q(n)). 4
 0, 1, 2, 4, 10, 32, 126, 588, 3170, 19384, 132550, 1002212, 8301930, 74767056, 727348814, 7601002876, 84920459890, 1010058659048, 12742908917718, 169962226236180, 2389587638934650, 35321010036943360, 547577222471444062 (list; graph; refs; listen; history; text; internal format)
 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, 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

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Last modified September 28 22:22 EDT 2023. Contains 365739 sequences. (Running on oeis4.)