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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
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
EXAMPLE
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
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_ /; n<rows] := Accumulate[T[n-1]];
A = Array[T, rows, 0] // Transpose;
F[n_ /; n<rows, m_ /; m<rows] := A[[n+1, m+1]];
Table[F[n-m, m], {n, 0, rows-1}, {m, 0, n}] (* Jean-François Alcover, Jun 29 2019 *)
PROG
(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
(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
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.
Sequence in context: A344824 A226173 A212633 * A248156 A352780 A331368
KEYWORD
nonn,tabl,easy
AUTHOR
Paul Curtz, Dec 16 2011
STATUS
approved

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Last modified April 20 12:24 EDT 2024. Contains 371843 sequences. (Running on oeis4.)