A185690 Exponential Riordan array (1,sin(x)).

1, 0, 1, -1, 0, 1, 0, -4, 0, 1, 1, 0, -10, 0, 1, 0, 16, 0, -20, 0, 1, -1, 0, 91, 0, -35, 0, 1, 0, -64, 0, 336, 0, -56, 0, 1, 1, 0, -820, 0, 966, 0, -84, 0, 1, 0, 256, 0, -5440, 0, 2352, 0, -120, 0, 1, -1, 0, 7381, 0, -24970, 0, 5082, 0, -165, 0, 1, 0, -1024, 0, 87296, 0, -90112, 0, 10032, 0, -220, 0, 1

Offset: 1

Author

Vladimir Kruchinin, Feb 10 2011

Keywords

• sign
• tabl

Comments

The row n=0 with T(0,0)=1 and the column T(n,0)=0, n>0, are not entered into the sequence here.

A signed version of A136630 (apart from row 0 and column 0). - Peter Bala, Oct 06 2011

Also the Bell transform of the sequence "a(n) = (-1)^(n/2) if n is even else 0" without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Examples

			Array begins:
   1;
   0,   1;
  -1,   0,   1;
   0,  -4,   0,   1;
   1,   0, -10,   0,   1;
   0,  16,   0, -20,   0,   1;
  -1,   0,  91,   0, -35,   0,   1;
   0, -64,   0, 336,   0, -56,   0,   1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565, [math.CO], 2010.

Crossrefs

Cf. A000111, A136630.

Programs

Maple

A185690 := proc(n,k) if type(k+n,'even') then 2^(1-k)/k! * add( (-1)^(floor((n+k)/2)-i)*binomial(k,i)*(2*i-k)^n,i=0..floor(k/2)) ; else 0; end if; end proc: # R. J. Mathar, Feb 21 2011
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::even,(-1)^(n/2),0), 10); # Peter Luschny, Jan 29 2016

Mathematica

t[n_, k_] /; OddQ[n - k] = 0; t[n_, k_] /; EvenQ[n - k] := 2^(1-k)/k!* Sum[ (-1)^(Floor[(n+k)/2] - i)*Binomial[k, i]*(2*i-k)^n, {i, 0, k/2}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, If[EvenQ[n], (-1)^(n/2), 0]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Python

from sympy import binomial, factorial as f, floor
def T(n, k):
    return 0 if (n - k)%2 else 2**(1 - k)*sum([(-1)**((n + k)//2 - i)*binomial(k, i)*(2*i - k)**n for i in range(k//2 + 1)])//f(k)
for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jul 11 2017

Formula

T(n,k) = 2^(1-k)/k! *Sum_{i=0..floor(k/2)} (-1)^(floor((n+k)/2)-i) *binomial(k,i) *(2*i-k)^n, for even(n-k).

Sum_{k=0..n} T(n+1,k+1)*k! = A000111(n). - Alexander Burstein, Aug 01 2025