This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A185690 #51 Aug 01 2025 19:03:36
%S A185690 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 A185690 -64,0,336,0,-56,0,1,1,0,-820,0,966,0,-84,0,1,0,256,0,-5440,0,2352,0,
%U A185690 -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 A185690 Exponential Riordan array (1,sin(x)).
%C A185690 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 A185690 A signed version of A136630 (apart from row 0 and column 0). - Peter Bala, Oct 06 2011
%C A185690 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 A185690 G. C. Greubel, <a href="/A185690/b185690.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185690 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565, [math.CO], 2010.
%F A185690 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).
%F A185690 Sum_{k=0..n} T(n+1,k+1)*k! = A000111(n). - _Alexander Burstein_, Aug 01 2025
%e A185690 Array begins:
%e A185690 1;
%e A185690 0, 1;
%e A185690 -1, 0, 1;
%e A185690 0, -4, 0, 1;
%e A185690 1, 0, -10, 0, 1;
%e A185690 0, 16, 0, -20, 0, 1;
%e A185690 -1, 0, 91, 0, -35, 0, 1;
%e A185690 0, -64, 0, 336, 0, -56, 0, 1;
%p A185690 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 A185690 # The function BellMatrix is defined in A264428.
%p A185690 # Adds (1,0,0,0, ..) as column 0.
%p A185690 BellMatrix(n -> `if`(n::even,(-1)^(n/2),0), 10); # _Peter Luschny_, Jan 29 2016
%t A185690 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 A185690 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A185690 rows = 12;
%t A185690 B = BellMatrix[Function[n, If[EvenQ[n], (-1)^(n/2), 0]], rows];
%t A185690 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A185690 (Python)
%o A185690 from sympy import binomial, factorial as f, floor
%o A185690 def T(n, k):
%o A185690 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 A185690 for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Jul 11 2017
%Y A185690 Cf. A000111, A136630.
%K A185690 sign,tabl
%O A185690 1,8
%A A185690 _Vladimir Kruchinin_, Feb 10 2011