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%I #55 Jan 01 2024 13:14:38
%S 1,2,3,16,30,15,272,588,420,105,7936,18960,16380,6300,945,353792,
%T 911328,893640,429660,103950,10395,22368256,61152000,65825760,
%U 36636600,11351340,1891890,135135,1903757312,5464904448,6327135360,3918554640,1427025600,310269960,37837800,2027025
%N 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).
%C A triangle related to Euler numbers and tangent numbers.
%C 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
%C 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
%H G. C. Greubel, <a href="/A085734/b085734.txt">Rows n = 0..100 of triangle, flattened</a>
%H Ghislain R. Franssens, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles </a>, JIS 9 (2006) 06.4.1.
%H Alan D. Sokal, <a href="https://arxiv.org/abs/1804.04498">The Euler and Springer numbers as moment sequences</a>, arXiv:1804.04498 [math.CO], 2018.
%H M. S. Tokmachev, <a href="https://vestnik.susu.ru/mmph/article/viewFile/8337/6806">Correlations Between Elements and Sequences in a Numerical Prism</a>, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.
%F T(n, k) = A083061(n, k)*2^(n-k). - _Philippe Deléham_, Feb 27 2005
%F E.g.f.: sec(x)^y. - _Vladeta Jovovic_, May 20 2007
%F 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
%e Triangle begins as:
%e 1;
%e 2, 3;
%e 16, 30, 15;
%e 272, 588, 420, 105; ...
%t 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 *)
%o (Maxima)
%o 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 */
%o (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)))};
%o for(n=0,5, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Mar 21 2019
%o (Sage)
%o @CachedFunction
%o def T(n,k):
%o if n==0 and k==0: return 1
%o 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)))
%o [[T(n, k) for k in (0..n)] for n in (0..7)] # _G. C. Greubel_, Mar 21 2019
%Y T(n, 0) = A000182(n), tangent numbers, T(n, n) = A001147(n+1), Sum_{k>=0} T(n, k) = A000364(n+1), Euler numbers.
%Y Cf. A088874.
%Y A subtriangle of A098906.
%K nonn,tabl,easy
%O 0,2
%A _Philippe Deléham_, Jul 20 2003
%E Edited and extended by _Ray Chandler_, Nov 23 2003
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