A185951 - OEIS

%I #30 Jan 22 2019 22:44:11

%S 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,

%T 896,0,2800,0,168,0,1,9,0,15372,0,8190,0,252,0,1,0,5760,0,114240,0,

%U 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

%N Exponential Riordan array (1, x*cosh(x)).

%C The column k=0 of the array (which contains T(0,0)=1 and otherwise zero) is not included in the triangle.

%C 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

%H G. C. Greubel, <a href="/A185951/b185951.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2013.

%F 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.

%e Array begins

%e    1,

%e    0,   1,

%e    3,   0,    1,

%e    0,  12,    0,    1,

%e    5,   0,   30,    0,    1,

%e    0,  120,   0,   60,    0,    1,

%e    7,   0,   735,   0,   105,   0,    1,

%e    0,  896,   0,  2800,   0,   168,   0,   1,

%e    9,   0,  15372,  0,  8190,   0,   252,  0,   1,

%e    0, 5760,   0, 114240,  0,  20160,  0,  360,  0,  1,

%e   11,   0, 270765,  0, 556710,  0,  43890, 0,  495, 0,  1,

%e    0, 33792,  0, 4118400, 0, 2084544, 0, 87120, 0, 660, 0, 1.

%p A185951 := proc(n,k)

%p    if n =k then

%p       1;

%p    else

%p       binomial(n,k)/2^k * add( binomial(k,i)*(k-2*i)^(n-k),i=0..k) ;

%p    end if;

%p end proc: # _R. J. Mathar_, Feb 22 2011

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> `if`(n::even,n+1,0), 10); # _Peter Luschny_, Jan 29 2016

%t 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 *)

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, If[EvenQ[n], n + 1, 0]], rows = 12];

%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%K nonn,tabl

%O 1,4

%A _Vladimir Kruchinin_, Feb 11 2011