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.

1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 148, 52, 10, 1, 945, 1333, 472, 96, 13, 1, 10395, 14664, 5197, 1066, 153, 16, 1, 135135, 190633, 67567, 13873, 2009, 223, 19, 1, 2027025, 2859496, 1013512, 208116, 30170, 3380, 306, 22, 1

Offset

0,4

Links

Table of n, a(n) for n=0..44.

Formula

T(n,n) = 1.

T(n,k) = binomial(n-1,k-1) + (2*n - 1) * T(n-1,k) for 0 < k < n.

Conjecture: M(n,k) = (-1)^(n-k) * T(n,k) is matrix inverse of A350512.

Example

Triangle T(n,k) for 0 <= k <= n starts:
n\k :        0        1        2       3      4     5    6   7  8
=================================================================
  0 :        1
  1 :        1        1
  2 :        3        4        1
  3 :       15       21        7       1
  4 :      105      148       52      10      1
  5 :      945     1333      472      96     13     1
  6 :    10395    14664     5197    1066    153    16    1
  7 :   135135   190633    67567   13873   2009   223   19   1
  8 :  2027025  2859496  1013512  208116  30170  3380  306  22  1
  etc.

Mathematica

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 *)

Crossrefs

Cf. A001147 (column 0), A286286 (column 1), A249349 (column 2).

Cf. A000007 (alternating row sums).

Cf. A350512.

Sequence in context: A100326 A303728 A321627 * A028338 A039757 A136228

Adjacent sequences: A350554 A350555 A350556 * A350558 A350559 A350560

Keyword

nonn,easy,tabl

Author

Werner Schulte, Jan 05 2022

Status

approved