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A085734 Triangle read by rows: T(0,0) = 1, T(n,k) = Sum_{j=max(0,1-k) to n-k} (2^j)*(binomial(k+j,1+j) + binomial(k+j+1,1+j))*T(n-1,k-1+j)). 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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.

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

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.

Sequence in context: A074759 A175699 A102882 * A302837 A034382 A034383

Adjacent sequences:  A085731 A085732 A085733 * A085735 A085736 A085737

KEYWORD

nonn,tabl,easy

AUTHOR

Philippe Deléham, Jul 20 2003

EXTENSIONS

Edited and extended by Ray Chandler Nov 23 2003

STATUS

approved

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Last modified June 17 22:17 EDT 2019. Contains 324200 sequences. (Running on oeis4.)