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.

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, -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, 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

Offset: 0

Author

Paul D. Hanna, Mar 30 2005

Keywords

• sign
• tabl

Comments

Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.

Riordan array ((1-2*x)/(1-x), x(1-2*x)/(1-x)). - Philippe Deléham, Dec 05 2015

Examples

			Triangle begins:
   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;
  -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; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1080

Crossrefs

Cf. A078011, A090132, A104968, A104969, A134824, A153881.

Cf. A347171 (rows reversed, up to signs).

Programs

Magma

A104967:= func< n,k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;
[A104967(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 09 2021

Maple

A104967:= (n,k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);
seq(seq( A104967(n,k), k=0..n), n=0..12); # G. C. Greubel, Jun 09 2021

Mathematica

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}]];
Table[T[n, k], {n, 13}, {k, n}]//Flatten (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

Maxima

T(n,k):=sum((-2)^i*binomial(k+1,i)*binomial(n-i,k),i,0,n-k); /* Vladimir Kruchinin, Nov 02 2011 */

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)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

Sage

def A104967_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A104967_row(n)) # Peter Luschny, Mar 16 2016

Formula

G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).

Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).

Sum_{k=0..n} abs(T(n, k)) = A104968(n).

Sum_{k=0..n} T(n, k)^2 = A104969(n).

T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - Vladimir Kruchinin, Nov 02 2011

Sum_{k=0..floor(n/2)} T(n-k, k) = A078011(n+2). - G. C. Greubel, Jun 09 2021