A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

1, 2, 1, 1, 3, 1, 0, 4, 4, 1, 0, 4, 8, 5, 1, 0, 4, 12, 13, 6, 1, 0, 4, 16, 25, 19, 7, 1, 0, 4, 20, 41, 44, 26, 8, 1, 0, 4, 24, 61, 85, 70, 34, 9, 1, 0, 4, 28, 85, 146, 155, 104, 43, 10, 1, 0, 4, 32, 113, 231, 301, 259, 147, 53, 11, 1, 0, 4, 36, 145, 344, 532, 560, 406, 200, 64, 12, 1

Offset: 0

Author

Paul Curtz, Dec 16 2011

Keywords

• nonn
• tabl
• easy

Comments

The array F(n,m), beginning with row n=0, is:

1, 1, 1, 1, 1, 1, 1,

2, 3, 4, 5, 6, 7, 8,

1, 4, 8, 13, 19, 26, 34,

0, 4, 12, 25, 44, 70, 104,

0, 4, 16, 41, 85, 155, 259,

0, 4, 20, 61, 146, 301, 560,

0, 4, 24, 85, 231, 532, 1092.

Columns after A130713, A113311, A008574 have signatures (3,-3,1), (4,-6,4,-1), (5,-10,10,-5,1), (6,-15,20,-15,6,-1) (from A135278(n+3)).

Inserting columns of zeros and pushing the columns down, plus alternating sign switches defines the following triangle T(n,2m) = (-1)^(m/2)*F(n-2m,m):

1,

2 0,

1 0 -1,

0 0 -3 0,

0 0 -4 0 1,

0 0 -4 0 4 0,

0 0 -4 0 8 0 -1

The row sums in the triangle are (-1)^n*A099838(n).

The companion to A201863 is

1

1 0

1 0 0

1 0 -2 0

1 0 -4 0 1

1 0 -6 0 5 0

1 0 -8 0 13 0 -2

1 0 -10 0 25 0 -12 0

1 0 -12 0 41 0 -38 0 4

1 0 -14 0 61 0 -88 0 28 0

1 0 -16 0 85 0 -170 0 104 0 -8

5th column: A001844; 7th column: -A035597=-2*A005900(n+1); 9th column: 4*A006325(n+2); 11th column: -8*(1,8,34,104) (from columns 4,5,6,7 of F(n,m)).

As a triangular array, this is the Riordan array ((1+x)^2, x/(1-x)). - Philippe Deléham, Feb 21 2012

Examples

			Triangle T(n,k) begins:
  1
  2, 1
  1, 3,  1
  0, 4,  4,  1
  0, 4,  8,  5,   1
  0, 4, 12, 13,   6,   1
  0, 4, 16, 25,  19,   7,   1
  0, 4, 20, 41,  44,  26,   8,  1
  0, 4, 24, 61,  85,  70,  34,  9,  1
  0, 4, 28, 85, 146, 155, 104, 43, 10, 1
- _Philippe Deléham_, Feb 21 2012
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5151

Crossrefs

Cf. A130713 (column 0), A113311 (column 1), A008574 (column 2), A001844 (column 3), A005900 (column 4), A006325 (column 5), A033455 (column 6).

Cf. A267633.

Programs

GAP

Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)+Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)-Binomial(n-3,n-k-3)))); # Muniru A Asiru, Mar 22 2018

Maple

A130713 := proc(n)
    if n <= 2 and n >=0 then
        op(n+1,[1,2,1]) ;
    else
        0;
    end if;
end proc:
A202241 := proc(n,m)
    option remember;
    if n < 0 then
        0 ;
    elif m = 0 then
        A130713(n);
    else
        procname(n,m-1)+procname(n-1,m) ;
    end if;
end proc:
for d from 0 to 12 do
    for m from 0 to d do
        printf("%d,",A202241(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Dec 22 2011
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
     seq(C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), k = 0..n);
end do; # Peter Bala, Mar 20 2018

Mathematica

rows = 12;
T[0] = PadRight[{1, 2, 1}, rows];
T[n_ /; nJean-François Alcover, Jun 29 2019 *)

Sage

def Trow(n): return [binomial(n, n-k) + binomial(n-1, n-k-1) - binomial(n-2, n-k-2) - binomial(n-3, n-k-3) for k in (0..n)]
for n in (0..9): print(Trow(n)) # Peter Luschny, Mar 21 2018

Formula

F(1,m) = m+2.

F(2,m) = A034856(m+1).

F(3,m) = A000297(m-1).

Sum_{m=0..d} F(d-m,m) = A116453(d-3), d >= 3 (antidiagonal sums).

As a triangular array T(n,k), 0 <= k <= n, satisfies: T(n,k) = T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 2, T(2,0) = 1, T(3,0) = 0. - Philippe Deléham, Feb 21 2012

Unsigned diagonals of A267633 (beginning with its main diagonal) appear to be the reverse rows of this entry's triangle beginning with the fourth row. - Tom Copeland, Jan 26 2016

T(n,k) = C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), where C(n, k) = n!/(k!*(n-k)!) if 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018