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A106195 Riordan array (1/(1-2x),x(1-x)/(1-2x)). 7
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, 128, 256, 272, 192, 96, 34, 8, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1024, 2816, 4096, 4048, 2972, 1683, 743, 253, 64, 11 (list; table; graph; refs; listen; history; text; internal format)
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 F(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),

where u(1,x)=1, v(1,x)=1.  Then v matches A106195 and u matches A207605.  (End)

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

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 *)

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

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 xrange(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 xrange(13): nprint ([T(n, k) for k in xrange(n + 1)]) # Indranil Ghosh, May 28 2017, after formula from Peter Luschny

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...

Cf. A055587, A007318, A001792, A002620, A049612, A029653, A078812, A208341.

Sequence in context: A103316 A140069 A105851 * A247023 A051129 A319075

Adjacent sequences:  A106192 A106193 A106194 * A106196 A106197 A106198

KEYWORD

easy,nonn,tabl

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

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)