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A085734
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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).
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4
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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Triangle begins as:
1;
2, 3;
16, 30, 15;
272, 588, 420, 105; ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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T(n, 0) = A000182(n), tangent numbers, T(n, n) = A001147(n+1), Sum_{k>=0} T(n, k) = A000364(n+1), Euler numbers.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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