|
%I #26 Dec 23 2021 05:52:33
%S 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,
%T 2269,781,261,85,25,9,1,1,8518,2920,976,316,100,28,10,1,1,32206,11006,
%U 3676,1196,376,116,31,11,1,1
%N Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.
%C The left factor of the matrix product is the triangle which starts
%C 1;
%C 2, 1;
%C 6, 3, 1;
%C 20, 10, 4, 1;
%C 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.
%C Swapping the two factors, A007318^(-1) * A046899(row-reversed) would generate A158793.
%C 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
%C 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
%C 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
%H G. C. Greubel, <a href="/A158815/b158815.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&filename=The%20Combinatorics%20of%20Convex%20Permutominoes.pdf">The Combinatorics of Convex Permutominoes</a>, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.
%F Sum_{k=0..n} T(n,k) = A046899(n).
%F T(n,0) = A026641(n).
%F 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
%F T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - _Peter Bala_, Jul 13 2021
%e The triangle starts
%e 1;
%e 1, 1;
%e 4, 1, 1;
%e 13, 5, 1, 1;
%e 46, 16, 6, 1, 1;
%e 166, 58, 19, 7, 1, 1;
%e 610, 211, 71, 22, 8, 1, 1;
%e 2269, 781, 261, 85, 25, 9, 1, 1;
%e 8518, 2620, 976, 316, 100, 28, 10, 1, 1;
%e 32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1;
%e 122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
%e ...
%p A158815 := proc (n, k)
%p add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
%p end proc:
%p seq(seq(A158815(n, k), k = 0..n), n = 0..10); # _Peter Bala_, Jul 13 2021
%t T[n_,k_]:= T[n,k]= Sum[(-1)^(j+k)*Binomial[j,k]*Binomial[2*n-j,n], {j,0,n}];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2021 *)
%o (Sage)
%o def A158815(n,k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
%o flatten([[A158815(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Dec 22 2021
%Y Cf. A000984, A026641, A046899, A158793, A171243.
%K nonn,tabl
%O 0,4
%A _Gary W. Adamson_ and _Roger L. Bagula_, Mar 27 2009
|
