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
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
Programs
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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
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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
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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 */
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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
Comments