OFFSET
0,2
COMMENTS
A triangle related to Euler numbers and tangent numbers.
T(n,k) = number of down-up permutations on [2n+2] with k+1 left-to-right maxima. For example, T(1,1) counts the following 3 down-up permutations on [4] each with 2 left-to-right maxima: 2143, 3142, 3241. - David Callan, Oct 25 2004
It appears that Sum_{k=0..n} (-1)^(n-k)*T(n,k)*x^(k+1) is the zeta polynomial for the poset of even-sized subsets of [2n+2] ordered by inclusion. - Geoffrey Critzer, Apr 22 2023
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles , JIS 9 (2006) 06.4.1.
Tian Han, Sergey Kitaev, and Philip B. Zhang, Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs, arXiv:2408.12865 [math.CO], 2024. See p. 4.
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.
FORMULA
T(n, k) = A083061(n, k)*2^(n-k). - Philippe Deléham, Feb 27 2005
E.g.f.: sec(x)^y. - Vladeta Jovovic, May 20 2007
T(n,m) = Sum_{k=1..n} (Stirling1(k,m)*Sum_{i=0..k-1} (i-k)^(2*n)* binomial(2*k,i)*(-1)^(n+m+i))/(2^(k-1)*k!). - Vladimir Kruchinin, May 20 2013
EXAMPLE
Triangle begins as:
1;
2, 3;
16, 30, 15;
272, 588, 420, 105; ...
MATHEMATICA
t[n_, k_]:= t[n, k] = Sum[(2^j)*(Binomial[k+j, 1+j] + Binomial[k+j+1, 1+j])*t[n-1, k-1+j], {j, Max[0, 1-k], n-k}]; t[0, 0] = 1; Table[t[n, k], {n, 0, 7}, {k, 0, n}]//Flatten (* Jean-François Alcover, Feb 26 2013 *)
PROG
(Maxima)
T(n, m):=sum((stirling1(k, m)*sum((i-k)^(2*n)*binomial(2*k, i)*(-1)^(n+m+i), i, 0, k-1))/(2^(k-1)*k!), k, 1, n); /* Vladimir Kruchinin, May 20 2013 */
(PARI) {T(n, k) = if(n==0 && k==0, 1, sum(j=max(0, 1-k), n-k, (2^j)*(binomial(k+j, 1+j) + binomial(k+j+1, 1+j))*T(n-1, k-1+j)))};
for(n=0, 5, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 21 2019
(Sage)
@CachedFunction
def T(n, k):
if n==0 and k==0: return 1
else: return sum((2^j)*(binomial(k+j, 1+j) + binomial(k+j+1, 1+j))*T(n-1, k-1+j) for j in (max(0, 1-k)..(n-k)))
[[T(n, k) for k in (0..n)] for n in (0..7)] # G. C. Greubel, Mar 21 2019
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Jul 20 2003
EXTENSIONS
Edited and extended by Ray Chandler, Nov 23 2003
STATUS
approved