OFFSET
0,4
COMMENTS
Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x). See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
1;
-1, 1;
-1, -1, 1;
1, -2, -1, 1;
3, 1, -3, -1, 1;
1, 6, 1, -4, -1, 1;
-7, 4, 10, 1, -5, -1, 1;
-13, -13, 8, 15, 1, -6, -1, 1;
3, -31, -23, 13, 21, 1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.
FORMULA
From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
EXAMPLE
n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
1;
1, 1;
2, 1, 1;
3, 3, 1, 1;
5, 5, 4, 1, 1;
8, 10, 7, 5, 1, 1;
13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
1;
0, 1;
0, 1, 1;
0, 2, 1, 1;
0, 3, 3, 1, 1;
0, 5, 5, 4, 1, 1;
0, 8, 10, 7, 5, 1, 1;
0, 13, 18, 16, 9, 6, 1, 1;
MAPLE
a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n, m), m=0..n), n=0..12); # Alois P. Heinz, Sep 19 2013
MATHEMATICA
Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k, 0, n}], {n, 0, 12}, {m, 0, n}]
PROG
(PARI) {T(n, k) = sum(j=0, n, binomial(j, k)*binomial(n-j, j-k))}; \\ G. C. Greubel, May 16 2019
(Magma) [[(&+[Binomial(j, k)*Binomial(n-j, j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019
(SageMath) [[sum(binomial(j, k)*binomial(n-j, j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> Binomial(j, k)*Binomial(n-j, j-k)) ))); # G. C. Greubel, May 16 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, May 23 2001
STATUS
approved
