OFFSET
0,4
COMMENTS
The left factor of the matrix product is the triangle which starts
1;
2, 1;
6, 3, 1;
20, 10, 4, 1;
a row-reversed version of A046899, equivalent to the triangular view of the array A092392. The right factor is the inverse of the matrix A007318, which is A130595.
Riordan array (f(x), g(x)) where f(x) is the g.f. of A026641 and where g(x) is the g.f. of A000957. - Philippe Deléham, Dec 05 2009
T(n,k) is the number of nonnegative paths consisting of upsteps U=(1,1) and downsteps D=(1,-1) of length 2n with k low peaks. (A low peak has its peak vertex at height 1.) Example: T(3,1)=5 counts UDUUUU, UDUUUD, UDUUDU, UDUUDD, UUDDUD. - David Callan, Nov 21 2011
Matrix product P^2 * Q * P^(-2), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A171243. - Peter Bala, Jul 13 2021
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.
FORMULA
Sum_{k=0..n} T(n,k) = A046899(n).
T(n,0) = A026641(n).
Sum_{k=0..n} T(n,k)*x^k = A026641(n), A000984(n), A001700(n), A000302(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - Peter Bala, Jul 13 2021
EXAMPLE
The triangle starts
1;
1, 1;
4, 1, 1;
13, 5, 1, 1;
46, 16, 6, 1, 1;
166, 58, 19, 7, 1, 1;
610, 211, 71, 22, 8, 1, 1;
2269, 781, 261, 85, 25, 9, 1, 1;
8518, 2620, 976, 316, 100, 28, 10, 1, 1;
32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1;
122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
...
MAPLE
A158815 := proc (n, k)
add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
end proc:
seq(seq(A158815(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021
MATHEMATICA
T[n_, k_]:= T[n, k]= Sum[(-1)^(j+k)*Binomial[j, k]*Binomial[2*n-j, n], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)
PROG
(Sage)
def A158815(n, k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
flatten([[A158815(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson and Roger L. Bagula, Mar 27 2009
STATUS
approved