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 A104967 #43 Jan 11 2025 18:10:28
%S A104967 1,-1,1,-1,-2,1,-1,-1,-3,1,-1,0,0,-4,1,-1,1,2,2,-5,1,-1,2,3,4,5,-6,1,
%T A104967 -1,3,3,3,5,9,-7,1,-1,4,2,0,0,4,14,-8,1,-1,5,0,-4,-6,-6,0,20,-9,1,-1,
%U A104967 6,-3,-8,-10,-12,-14,-8,27,-10,1,-1,7,-7,-11,-10,-10,-14,-22,-21,35,-11,1,-1,8,-12,-12,-5,0,0,-8,-27,-40,44,-12,1
%N A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.
%C A104967 Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.
%C A104967 Riordan array ((1-2*x)/(1-x), x(1-2*x)/(1-x)). - _Philippe Deléham_, Dec 05 2015
%H A104967 Paul D. Hanna, <a href="/A104967/b104967.txt">Table of n, a(n) for n = 0..1080</a>
%F A104967 G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
%F A104967 Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).
%F A104967 Sum_{k=0..n} abs(T(n, k)) = A104968(n).
%F A104967 Sum_{k=0..n} T(n, k)^2 = A104969(n).
%F A104967 T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - _Vladimir Kruchinin_, Nov 02 2011
%F A104967 Sum_{k=0..floor(n/2)} T(n-k, k) = A078011(n+2). - _G. C. Greubel_, Jun 09 2021
%e A104967 Triangle begins:
%e A104967 1;
%e A104967 -1, 1;
%e A104967 -1, -2, 1;
%e A104967 -1, -1, -3, 1;
%e A104967 -1, 0, 0, -4, 1;
%e A104967 -1, 1, 2, 2, -5, 1;
%e A104967 -1, 2, 3, 4, 5, -6, 1;
%e A104967 -1, 3, 3, 3, 5, 9, -7, 1;
%e A104967 -1, 4, 2, 0, 0, 4, 14, -8, 1;
%e A104967 -1, 5, 0, -4, -6, -6, 0, 20, -9, 1; ...
%p A104967 A104967:= (n,k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);
%p A104967 seq(seq( A104967(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Jun 09 2021
%t A104967 T[n_, k_]:= T[n, k]= Which[k==n, 1, k==0, 0, True, T[n-1, k-1] - Sum[T[n-i, k-1], {i, 2, n-k+1}]];
%t A104967 Table[T[n, k], {n, 13}, {k, n}]//Flatten (* _Jean-François Alcover_, Jun 11 2019, after _Peter Luschny_ *)
%o A104967 (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)),n,x),k,y)}
%o A104967 for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))
%o A104967 (Maxima) T(n,k):=sum((-2)^i*binomial(k+1,i)*binomial(n-i,k),i,0,n-k); /* _Vladimir Kruchinin_, Nov 02 2011 */
%o A104967 (Sage)
%o A104967 def A104967_row(n):
%o A104967 @cached_function
%o A104967 def prec(n, k):
%o A104967 if k==n: return 1
%o A104967 if k==0: return 0
%o A104967 return prec(n-1,k-1)-sum(prec(n-i,k-1) for i in (2..n-k+1))
%o A104967 return [prec(n, k) for k in (1..n)]
%o A104967 for n in (1..10): print(A104967_row(n)) # _Peter Luschny_, Mar 16 2016
%o A104967 (Magma)
%o A104967 A104967:= func< n,k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;
%o A104967 [A104967(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 09 2021
%Y A104967 Cf. A078011, A090132, A104968, A104969, A134824, A153881.
%Y A104967 Cf. A347171 (rows reversed, up to signs).
%K A104967 sign,tabl
%O A104967 0,5
%A A104967 _Paul D. Hanna_, Mar 30 2005