A085734 Triangle read by rows: T(0,0) = 1, T(n,k) = 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).

1, 2, 3, 16, 30, 15, 272, 588, 420, 105, 7936, 18960, 16380, 6300, 945, 353792, 911328, 893640, 429660, 103950, 10395, 22368256, 61152000, 65825760, 36636600, 11351340, 1891890, 135135, 1903757312, 5464904448, 6327135360, 3918554640, 1427025600, 310269960, 37837800, 2027025

Offset: 0

Author

Philippe Deléham, Jul 20 2003

Keywords

• nonn
• tabl
• easy

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

Examples

			Triangle begins as:
    1;
    2,   3;
   16,  30,  15;
  272, 588, 420, 105; ...
		

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.

Crossrefs

T(n, 0) = A000182(n), tangent numbers, T(n, n) = A001147(n+1), Sum_{k>=0} T(n, k) = A000364(n+1), Euler numbers.

Cf. A088874.

A subtriangle of A098906.

Programs

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 *)

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

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

Extensions

Edited and extended by Ray Chandler, Nov 23 2003