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%I #62 Jun 29 2019 03:55:28
%S 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,
%T 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,
%U 4,32,113,231,301,259,147,53,11,1,0,4,36,145,344,532,560,406,200,64,12,1
%N 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.
%C The array F(n,m), beginning with row n=0, is:
%C 1, 1, 1, 1, 1, 1, 1,
%C 2, 3, 4, 5, 6, 7, 8,
%C 1, 4, 8, 13, 19, 26, 34,
%C 0, 4, 12, 25, 44, 70, 104,
%C 0, 4, 16, 41, 85, 155, 259,
%C 0, 4, 20, 61, 146, 301, 560,
%C 0, 4, 24, 85, 231, 532, 1092.
%C 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)).
%C 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):
%C 1,
%C 2 0,
%C 1 0 -1,
%C 0 0 -3 0,
%C 0 0 -4 0 1,
%C 0 0 -4 0 4 0,
%C 0 0 -4 0 8 0 -1
%C The row sums in the triangle are (-1)^n*A099838(n).
%C The companion to A201863 is
%C 1
%C 1 0
%C 1 0 0
%C 1 0 -2 0
%C 1 0 -4 0 1
%C 1 0 -6 0 5 0
%C 1 0 -8 0 13 0 -2
%C 1 0 -10 0 25 0 -12 0
%C 1 0 -12 0 41 0 -38 0 4
%C 1 0 -14 0 61 0 -88 0 28 0
%C 1 0 -16 0 85 0 -170 0 104 0 -8
%C 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)).
%C As a triangular array, this is the Riordan array ((1+x)^2, x/(1-x)). - _Philippe Deléham_, Feb 21 2012
%H Muniru A Asiru, <a href="/A202241/b202241.txt">Table of n, a(n) for n = 0..5151</a>
%F F(1,m) = m+2.
%F F(2,m) = A034856(m+1).
%F F(3,m) = A000297(m-1).
%F Sum_{m=0..d} F(d-m,m) = A116453(d-3), d >= 3 (antidiagonal sums).
%F 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
%F 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
%F 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
%e Triangle T(n,k) begins:
%e 1
%e 2, 1
%e 1, 3, 1
%e 0, 4, 4, 1
%e 0, 4, 8, 5, 1
%e 0, 4, 12, 13, 6, 1
%e 0, 4, 16, 25, 19, 7, 1
%e 0, 4, 20, 41, 44, 26, 8, 1
%e 0, 4, 24, 61, 85, 70, 34, 9, 1
%e 0, 4, 28, 85, 146, 155, 104, 43, 10, 1
%e - _Philippe Deléham_, Feb 21 2012
%p A130713 := proc(n)
%p if n <= 2 and n >=0 then
%p op(n+1,[1,2,1]) ;
%p else
%p 0;
%p end if;
%p end proc:
%p A202241 := proc(n,m)
%p option remember;
%p if n < 0 then
%p 0 ;
%p elif m = 0 then
%p A130713(n);
%p else
%p procname(n,m-1)+procname(n-1,m) ;
%p end if;
%p end proc:
%p for d from 0 to 12 do
%p for m from 0 to d do
%p printf("%d,",A202241(d-m,m)) ;
%p end do:
%p end do: # _R. J. Mathar_, Dec 22 2011
%p C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
%p for n from 0 to 10 do
%p 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);
%p end do; # _Peter Bala_, Mar 20 2018
%t rows = 12;
%t T[0] = PadRight[{1, 2, 1}, rows];
%t T[n_ /; n<rows] := Accumulate[T[n-1]];
%t A = Array[T, rows, 0] // Transpose;
%t F[n_ /; n<rows, m_ /; m<rows] := A[[n+1, m+1]];
%t Table[F[n-m, m], {n, 0, rows-1}, {m, 0, n}] (* _Jean-François Alcover_, Jun 29 2019 *)
%o (Sage)
%o 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)]
%o for n in (0..9): print(Trow(n)) # _Peter Luschny_, Mar 21 2018
%o (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
%Y Cf. A130713 (column 0), A113311 (column 1), A008574 (column 2), A001844 (column 3), A005900 (column 4), A006325 (column 5), A033455 (column 6).
%Y Cf. A267633.
%K nonn,tabl,easy
%O 0,2
%A _Paul Curtz_, Dec 16 2011
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