cp's OEIS Frontend

A350557 Triangle T(n,k) read by rows with T(n,0) = (2*n)! / (2^n * n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2*i)!) * (2*n)! / (2^n * n!) for 0 < k <= n.

%I A350557 #8 Jan 18 2022 21:46:34
%S A350557 1,1,1,3,4,1,15,21,7,1,105,148,52,10,1,945,1333,472,96,13,1,10395,
%T A350557 14664,5197,1066,153,16,1,135135,190633,67567,13873,2009,223,19,1,
%U A350557 2027025,2859496,1013512,208116,30170,3380,306,22,1
%N A350557 Triangle T(n,k) read by rows with T(n,0) = (2*n)! / (2^n * n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2*i)!) * (2*n)! / (2^n * n!) for 0 < k <= n.
%F A350557 T(n,n) = 1.
%F A350557 T(n,k) = binomial(n-1,k-1) + (2*n - 1) * T(n-1,k) for 0 < k < n.
%F A350557 Conjecture: M(n,k) = (-1)^(n-k) * T(n,k) is matrix inverse of A350512.
%e A350557 Triangle T(n,k) for 0 <= k <= n starts:
%e A350557 n\k :        0        1        2       3      4     5    6   7  8
%e A350557 =================================================================
%e A350557   0 :        1
%e A350557   1 :        1        1
%e A350557   2 :        3        4        1
%e A350557   3 :       15       21        7       1
%e A350557   4 :      105      148       52      10      1
%e A350557   5 :      945     1333      472      96     13     1
%e A350557   6 :    10395    14664     5197    1066    153    16    1
%e A350557   7 :   135135   190633    67567   13873   2009   223   19   1
%e A350557   8 :  2027025  2859496  1013512  208116  30170  3380  306  22  1
%e A350557   etc.
%t A350557 Flatten[Table[If[k==0,(2n)!/(2^n n!),Sum[Binomial[i-1,k-1]2^i i!/(2i)!,{i,k,n}](2n)!/(2^n n!)],{n,0,8},{k,0,n}]] (* _Stefano Spezia_, Jan 06 2022 *)
%Y A350557 Cf. A001147 (column 0), A286286 (column 1), A249349 (column 2).
%Y A350557 Cf. A000007 (alternating row sums).
%Y A350557 Cf. A350512.
%K A350557 nonn,easy,tabl
%O A350557 0,4
%A A350557 _Werner Schulte_, Jan 05 2022