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Exponential Riordan array (1,sin(x)).
2

%I #42 May 20 2021 12:46:39

%S 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,

%T -64,0,336,0,-56,0,1,1,0,-820,0,966,0,-84,0,1,0,256,0,-5440,0,2352,0,

%U -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

%N Exponential Riordan array (1,sin(x)).

%C 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.

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

%C 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

%H G. C. Greubel, <a href="/A185690/b185690.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565, [math.CO], 2010.

%F 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).

%e Array begins:

%e 1;

%e 0, 1;

%e -1, 0, 1;

%e 0, -4, 0, 1;

%e 1, 0, -10, 0, 1;

%e 0, 16, 0, -20, 0, 1;

%e -1, 0, 91, 0, -35, 0, 1;

%e 0, -64, 0, 336, 0, -56, 0, 1;

%p 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

%p # The function BellMatrix is defined in A264428.

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

%p BellMatrix(n -> `if`(n::even,(-1)^(n/2),0), 10); # _Peter Luschny_, Jan 29 2016

%t 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 *)

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

%t rows = 12;

%t B = BellMatrix[Function[n, If[EvenQ[n], (-1)^(n/2), 0]], rows];

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

%o (Python)

%o from sympy import binomial, factorial as f, floor

%o def T(n, k):

%o 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)

%o for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Jul 11 2017

%Y Cf. A136630.

%K sign,tabl

%O 1,8

%A _Vladimir Kruchinin_, Feb 10 2011