OFFSET
0,2
COMMENTS
Extract antidiagonals from the product P * A, where P = the infinite lower triangular Pascal's triangle matrix; and A = the Pascal's triangle array:
1, 1, 1, 1, ...
1, 2, 3, 4, ...
1, 3, 6, 10, ...
1, 4, 10, 20, ...
...
Row sums are Fibonacci(2n+2). Diagonal sums are A006054(n+2). Row sums of inverse are A105523. Product of Pascal triangle A007318 and A046854.
A106195 with an appended column of ones = A055587. Alternatively, k-th column (k=0, 1, 2) is the binomial transform of bin(n, k).
T(n,k) is the number of ideals in the fence Z(2n) having k elements of rank 1. - Emanuele Munarini, Mar 22 2011
Subtriangle of the triangle given by (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
FORMULA
T(n,k) = Sum_{j=0..n} C(n-k,n-j)*C(j,k).
From Emanuele Munarini, Mar 22 2011: (Start)
T(n,k) = Sum_{i=0..n-k} C(k,i)*C(n-k,i)*2^(n-k-i).
T(n,k) = Sum_{i=0..n-k} C(k,i)*C(n-i,k)*(-1)^i*2^(n-k-i).
Recurrence: T(n+2,k+1) = 2*T(n+1,k+1)+T(n+1,k)-T(n,k) (End)
From Clark Kimberling, Feb 19 2012: Define
u(n,x) = u(n-1,x)+v(n-1,x), v(n,x) = u(n-1,x)+(x+1)v(n-1,x),
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1). - Philippe Deléham, Mar 22 2012
T(n+k,k) is the coefficient of x^n y^k in 1/(1-2x-y+xy). - Ira M. Gessel, Oct 30 2012
T(n, k) = A208341(n+1,n-k+1), k = 0..n. - Reinhard Zumkeller, Dec 16 2013
T(n, k) = hypergeometric_2F1(-n+k, k+1, 1 , -1). - Peter Luschny, May 20 2015
G.f. 1/(1-2*x+x^2*y-x*y). - R. J. Mathar, Aug 11 2015
Sum_{k=0..n} T(n, k) = Fibonacci(2*n+2) = A088305(n+1). - G. C. Greubel, Mar 15 2020
EXAMPLE
Triangle begins
1;
2, 1;
4, 3, 1;
8, 8, 4, 1;
16, 20, 13, 5, 1;
32, 48, 38, 19, 6, 1;
64, 112, 104, 63, 26, 7, 1;
(0, 2, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...) begins :
1;
0, 1;
0, 2, 1;
0, 4, 3, 1;
0, 8, 8, 4, 1;
0, 16, 20, 13, 5, 1;
0, 32, 48, 38, 19, 6, 1;
0, 64, 112, 104, 63, 26, 7, 1. - Philippe Deléham, Mar 22 2012
MAPLE
T := (n, k) -> hypergeom([-n+k, k+1], [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 = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A207605 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A106195 *)
(* Clark Kimberling, Feb 19 2012 *)
Table[Hypergeometric2F1[-n+k, k+1, 1, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 15 2020 *)
PROG
(Maxima) create_list(sum(binomial(i, k)*binomial(n-k, n-i), i, 0, n), n, 0, 8, k, 0, n); [Emanuele Munarini, Mar 22 2011]
(Haskell)
a106195 n k = a106195_tabl !! n !! k
a106195_row n = a106195_tabl !! n
a106195_tabl = [1] : [2, 1] : f [1] [2, 1] where
f us vs = ws : f vs ws where
ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
([0] ++ us ++ [0])
-- Reinhard Zumkeller, Dec 16 2013
(Python)
from sympy import Poly, symbols
x = symbols('x')
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017
(Python)
from mpmath import hyp2f1, nprint
def T(n, k): return hyp2f1(k - n, k + 1, 1, -1)
for n in range(13): nprint([int(T(n, k)) for k in range(n + 1)]) # Indranil Ghosh, May 28 2017, after formula from Peter Luschny
(Magma) [ (&+[Binomial(n-k, n-j)*Binomial(j, k): j in [0..n]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 15 2020
(Sage) [[sum(binomial(n-k, n-j)*binomial(j, k) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 15 2020
CROSSREFS
Column 0 = 1, 2, 4...; (binomial transform of 1, 1, 1...); column 1 = 1, 3, 8, 20...(binomial transform of 1, 2, 3...); column 2: 1, 4, 13, 38...= binomial transform of bin(n, 2): 1, 3, 6...
KEYWORD
AUTHOR
Gary W. Adamson, Apr 24 2005; Paul Barry, May 21 2006
EXTENSIONS
Edited by N. J. A. Sloane, Apr 09 2007, merging two sequences submitted independently by Gary W. Adamson and Paul Barry
STATUS
approved