OFFSET
0,2
COMMENTS
Riordan array (c(x^2)/(1-2xc(x^2)),xc(x^2)) where c(x)=g.f. of Catalan numbers A000108. - Philippe Deléham, Mar 18 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
Sum_{k=0..n} T(n,k)*(-k+1) = 2^n. - Philippe Deléham, Mar 25 2007
From Peter Bala, Feb 20 2018: (Start)
T(n,k) = Sum_{j = 0..floor((n-k)/2)} 2^(n-k-2*j)*binomial(n, j) - Sum_{j = 0..floor((n-k-2)/2)} 2^(n-k-2-2*j)*binomial(n, j), 0 <= k <= n. - Peter Bala, Feb 20 2018
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 - x^2)/(1 - 2*x) * (1 + x^2)^n about 0. For example, for n = 4, (1 - x^2)/(1 - 2*x) * (1 + x^2)^4 = (30*x^4 + 14*x*3 + 7*x^2 + 2*x + 1) + O(x^5). (End)
EXAMPLE
Triangle begins:
1;
2, 1;
5, 2, 1;
12, 6, 2, 1;
30, 14, 7, 2, 1;
74, 37, 16, 8, 2, 1;
185, 90, 45, 18, 9, 2, 1;
460, 230, 108, 54, 20, 10, 2, 1;
1150, 568, 284, 128, 64, 22, 11, 2, 1;
2868, 1434, 696, 348, 150, 75, 24, 12, 2, 1;
MAPLE
A126075 := proc (n, k)
add( 2^(n-k-2*j)*binomial(n, j), j = 0..floor((n-k)/2) ) - add( 2^(n-k-2-2*j)*binomial(n, j), j = 0..floor((n-k-2)/2) )
end proc:
# display sequence in triangular form
for n from 0 to 10 do seq(A126075(n, k), k = 0..n) end do;
# Peter Bala, Feb 20 2018
MATHEMATICA
T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 2, 0], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Mar 02 2007
STATUS
approved