OFFSET
1,4
COMMENTS
The column k=0 of the array (which contains T(0,0)=1 and otherwise zero) is not included in the triangle.
Also the Bell transform of the sequence "a(n) = n+1 if n is even else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
T(n,k) = binomial(n,k)/(2^k) * Sum_{i=0..k} binomial(k,i) *(k-2*i)^(n-k), n > k; T(n,n) = 1.
EXAMPLE
Array begins
1,
0, 1,
3, 0, 1,
0, 12, 0, 1,
5, 0, 30, 0, 1,
0, 120, 0, 60, 0, 1,
7, 0, 735, 0, 105, 0, 1,
0, 896, 0, 2800, 0, 168, 0, 1,
9, 0, 15372, 0, 8190, 0, 252, 0, 1,
0, 5760, 0, 114240, 0, 20160, 0, 360, 0, 1,
11, 0, 270765, 0, 556710, 0, 43890, 0, 495, 0, 1,
0, 33792, 0, 4118400, 0, 2084544, 0, 87120, 0, 660, 0, 1.
MAPLE
A185951 := proc(n, k)
if n =k then
1;
else
binomial(n, k)/2^k * add( binomial(k, i)*(k-2*i)^(n-k), i=0..k) ;
end if;
end proc: # R. J. Mathar, Feb 22 2011
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n::even, n+1, 0), 10); # Peter Luschny, Jan 29 2016
MATHEMATICA
t[n_, k_] := Binomial[n, k]/(2^k)* Sum[ Binomial[k, i]*(k-2*i)^(n-k), {i, 0, k}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 14 2013, from formula *)
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], n + 1, 0]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
CROSSREFS
KEYWORD
AUTHOR
Vladimir Kruchinin, Feb 11 2011
STATUS
approved