A188832 Triangle T(n,k) = coefficient of x^n/n! in expansion of (x/cos x)^k/k!.

1, 0, 1, 3, 0, 1, 0, 12, 0, 1, 25, 0, 30, 0, 1, 0, 240, 0, 60, 0, 1, 427, 0, 1155, 0, 105, 0, 1, 0, 7616, 0, 3920, 0, 168, 0, 1, 12465, 0, 60732, 0, 10710, 0, 252, 0, 1, 0, 357120, 0, 315840, 0, 25200, 0, 360, 0, 1

Offset

1,4

Comments

Exponential Riordan array (1,x/cos(x)). The column of index 0 contains a 1 followed by zeros and is not reproduced in this triangle.

Links

Table of n, a(n) for n=1..55.

Formula

T(n,k)=binomial(n,k)*(1+(-1)^(n-k))*sum(m=1..n-k , (-1)^m*binomial(m+k-1,k-1)*sum(j=1..m, 2^(-j)*(sum(i=0..floor((j-1)/2)) , (j-2*i)^(n-k)*binomial(j,i))*binomial(m,j)*(-1)^((n-k)/2-m+j))), n>k, T(n,n)=1.

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n::even, (-1)^(n/2)*(n+1)*euler(n), 0), 10); # Peter Luschny, Jan 29 2016

Mathematica

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[EvenQ[n], (-1)^(n/2)*(n + 1)*EulerE[n], 0]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Prog

(Maxima)
T(n, k):=if n=k then 1 else binomial(n, k)*(1+(-1)^(n-k))*sum((-1)^m*binomial(m+k-1, k-1)*sum(2^(-j)*(sum((j-2*i)^(n-k)*binomial(j, i), i, 0, floor((j-1)/2)))*binomial(m, j)*(-1)^((n-k)/2-m+j), j, 1, m), m, 1, n-k);

Crossrefs

Sequence in context: A319234 A210473 A185951 * A279514 A094675 A307808

Adjacent sequences: A188829 A188830 A188831 * A188833 A188834 A188835

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Apr 11 2011

Status

approved