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A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.

%I A158793 #16 Jul 21 2021 10:02:19
%S A158793 1,1,1,3,1,1,7,4,1,1,19,9,5,1,1,51,26,11,6,1,1,141,70,34,13,7,1,1,393,
%T A158793 197,92,43,15,8,1,1,1107,553,265,117,53,17,9,1,1,3139,1570,751,346,
%U A158793 145,64,19,10,1,1,8953,4476,2156,991,441,176,76,21,11,1,1
%N A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.
%C A158793 Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A002426 and where g(x) is the g.f. of A005043. - _Philippe Deléham_, Dec 05 2009
%C A158793 Matrix product P * Q * P^(-1), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158815 and A171243. - _Peter Bala_, Jul 13 2021
%F A158793 T(n, m) = Sum_{k=m..n-1} A130595(n,k) * A092392(k+1,m+1), with the triangular interpretation of A092392.
%F A158793 Conjecture: T(n,1) = A113682(n-1). - _R. J. Mathar_, Oct 06 2009
%F A158793 Sum_{k=0..n} T(n,k)*x^k = A002426(n), A005773(n+1), A000244(n), A126932(n) for x = 0,1,2,3 respectively. - _Philippe Deléham_, Dec 03 2009
%F A158793 T(n, k) = (-1)^(k + n) binomial(n, k) hypergeom([k/2 + 1/2, k/2 + 1, k - n], [k + 1, k + 1], 4). - _Peter Luschny_, Jul 17 2021
%e A158793 First rows of the triangle:
%e A158793      1;
%e A158793      1,    1;
%e A158793      3,    1,    1;
%e A158793      7,    4,    1,   1;
%e A158793     19,    9,    5,   1,   1;
%e A158793     51,   26,   11,   6,   1,   1;
%e A158793    141,   70,   34,  13,   7,   1,  1;
%e A158793    393,  197,   92,  43,  15,   8,  1,  1;
%e A158793   1107,  553,  265, 117,  53,  17,  9,  1,  1;
%e A158793   3139, 1570,  751, 346, 145,  64, 19, 10,  1, 1;
%e A158793   8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1;
%p A158793 A158793 := proc (n, k)
%p A158793   add((-1)^(n+j)*binomial(n, j)*binomial(2*j-k, j-k), j = k..n);
%p A158793 end proc:
%p A158793 seq(seq(A158793(n, k), k = 0..n), n = 0..10); # _Peter Bala_, Jul 13 2021
%t A158793 T[n_, k_] := (-1)^(k + n) Binomial[n, k] HypergeometricPFQ[{k/2 + 1/2, k/2 + 1, k - n}, {k + 1, k + 1}, 4];
%t A158793 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Peter Luschny_, Jul 17 2021 *)
%Y A158793 T(n, 0) = A002426(n), A005773 (row sums).
%Y A158793 Cf. A046899, A007318, A158815, A171243.
%K A158793 nonn,tabl
%O A158793 0,4
%A A158793 _Gary W. Adamson_ & _Roger L. Bagula_, Mar 26 2009
%E A158793 Simplified definition from _R. J. Mathar_, Oct 06 2009