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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.

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