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A106195 Riordan array (1/(1-2*x), x*(1-x)/(1-2*x)). 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 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),

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

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...

Cf. A001792, A001906, A002620, A007318, A029653, A049612, A055587, A078812, A208341.

Sequence in context: A332389 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 August 5 08:27 EDT 2021. Contains 346464 sequences. (Running on oeis4.)