cp's OEIS Frontend

A120257 Triangle of Hankel transforms of certain binomial sums.

%I A120257 #15 Jan 13 2022 06:15:50
%S A120257 1,2,-1,3,-6,-1,4,-20,-20,1,5,-50,-175,70,1,6,-105,-980,1764,252,-1,7,
%T A120257 -196,-4116,24696,19404,-924,-1,8,-336,-14112,232848,731808,-226512,
%U A120257 -3432,1,9,-540,-41580,1646568,16818516,-24293412,-2760615,12870,1,10,-825,-108900,9343620,267227532,-1447482465
%N A120257 Triangle of Hankel transforms of certain binomial sums.
%C A120257 Row k is the Hankel transform of Sum_{j=0..n} binomial(k+j, j). Absolute value is reversal of A103905. Diagonal and subdiagonals are essentially signed versions of the central coefficients of certain generalized Pascal-Narayana triangles (A007318, A001263, A056939, A056940, A056941).
%F A120257 T(n, k) = (cos(Pi*k/2) - sin(Pi*k/2)) * Product_{j=0..n-k-1} C(2k+2+j, k+1)/C(k+1+j, j).
%e A120257 Triangle begins
%e A120257   1;
%e A120257   2,   -1;
%e A120257   3,   -6,     -1;
%e A120257   4,  -20,    -20,      1;
%e A120257   5,  -50,   -175,     70,      1;
%e A120257   6, -105,   -980,   1764,    252,      -1;
%e A120257   7, -196,  -4116,  24696,  19404,    -924,    -1;
%e A120257   8, -336, -14112, 232848, 731808, -226512, -3432, 1;
%o A120257 (PARI) T(n, k) = (-1)^((k+1)\2) * prod(j=0, n-k-1, binomial(2*k+2+j, k+1)/binomial(k+1+j, j)); \\ _Michel Marcus_, Jan 13 2022
%Y A120257 Cf. A120258.
%K A120257 easy,sign,tabl
%O A120257 0,2
%A A120257 _Paul Barry_, Jun 13 2006