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A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.

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