A185951 Exponential Riordan array (1, x*cosh(x)).

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

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, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 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

Sequence in context: A132884 A319234 A210473 * A188832 A279514 A094675

Adjacent sequences: A185948 A185949 A185950 * A185952 A185953 A185954

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 11 2011

Status

approved