OFFSET
0,3
COMMENTS
Previous name was: Triangle of coefficients of polynomials v(n,x) jointly generated with A160232; see the Formula section.
Row sums: (1,3,8,...), even-indexed Fibonacci numbers.
Alt. row sums: (1,-1,2,-3,...), signed Fibonacci numbers.
Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012
Essentially triangle in A049600. - Philippe Deléham, Mar 23 2012
LINKS
Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1.
As DELTA-triangle with 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 12 2012
G.f.: (1-2*y*x+y*x^2)/(1-x-2*y*x+y*x^2). - Philippe Deléham, Mar 12 2012
T(n,k) = A106195(n-1,n-k), k = 1..n. - Reinhard Zumkeller, Dec 16 2013
From Peter Bala, Aug 11 2015: (Start)
The following remarks assume the row and column indexing start at 0.
T(n,k) = Sum_{i = 0..k} 2^(k-i)*binomial(n-k,i)*binomial(k,i) = Sum_{i = 0..k} binomial(n-k+i,i)*binomial(k,i).
Riordan array (1/(1 - x), x*(2 - x)/(1 - x)).
O.g.f. 1/(1 - (2*t + 1)*x + t*x^2) = 1 + (1 + 2*t)*x + (1 + 3*t + 4*t^2)*x^2 + ....
Read as a square array, this equals P * transpose(P^2), where P denotes Pascal's triangle A007318. (End)
For k<n, T(n,k) = T(n-1,k) + Sum_{i=1..k} T(n-i,k-i). - Glen Whitney, Aug 17 2021
EXAMPLE
First five rows:
1;
1, 2;
1, 3, 4;
1, 4, 8, 8;
1, 5, 13, 20, 16;
First five polynomials v(n,x):
1
1 + 2x
1 + 3x + 4x^2
1 + 4x + 8x^2 + 8x^3
1 + 5x + 13x^2 + 20x^3 + 16x^4
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 3, 4, 0;
1, 4, 8, 8, 0;
1, 5, 13, 20, 16, 0;
1, 6, 19, 38, 48, 32, 0;
Triangle in A049600 begins:
0;
0, 1;
0, 1, 2;
0, 1, 3, 4;
0, 1, 4, 8, 8;
0, 1, 5, 13, 20, 16;
0, 1, 6, 19, 38, 48, 32;
... - Philippe Deléham, Mar 23 2012
MAPLE
T := (n, k) -> hypergeom([n-k+1, -k], [1], -1):
seq(lprint(seq(simplify(T(n, k)), k=0..n)), n=0..7); # Peter Luschny, May 20 2015
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208341 *)
PROG
(Haskell)
a208341 n k = a208341_tabl !! (n-1) !! (k-1)
a208341_row n = a208341_tabl !! (n-1)
a208341_tabl = map reverse a106195_tabl
-- Reinhard Zumkeller, Dec 16 2013
(PARI) T(n, k) = sum(i = 0, k, 2^(k-i)*binomial(n-k, i)*binomial(k, i));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Aug 14 2015
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 25 2012
EXTENSIONS
New name from Peter Luschny, May 20 2015
Offset corrected by Joerg Arndt, Aug 12 2015
STATUS
approved