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

Author

Vladimir Kruchinin, Feb 11 2011

Keywords

• nonn
• tabl

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

Examples

			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.
		

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.

Crossrefs

Cf. A264428.

Programs

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 *)

PARI

T(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Feb 25 2025

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.