A321630 The Riordan square of the Euler numbers. Triangle T(n, k), 0 <= k <= n, read by rows.

1, 1, 1, 5, 6, 1, 61, 71, 11, 1, 1385, 1532, 162, 16, 1, 50521, 53901, 3638, 278, 21, 1, 2702765, 2821378, 124723, 6504, 419, 26, 1, 199360981, 205440691, 6288561, 218551, 10255, 585, 31, 1, 19391512145, 19825343544, 444121924, 10616960, 340710, 15016, 776, 36, 1

Offset

0,4

Comments

The Riordan square is defined in A321620.

Triangle, read by rows, given by [1, 4, 9, 16, 25, 36, 49, …] DELTA [1, 0, 0, 0, 0, 0, 0, 0, …] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2020

Links

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

Example

[0] 1;
[1] 1,           1;
[2] 5,           6,           1;
[3] 61,          71,          11,        1;
[4] 1385,        1532,        162,       16,       1;
[5] 50521,       53901,       3638,      278,      21,     1;
[6] 2702765,     2821378,     124723,    6504,     419,    26,    1;
[7] 199360981,   205440691,   6288561,   218551,   10255,  585,   31,  1;
[8] 19391512145, 19825343544, 444121924, 10616960, 340710, 15016, 776, 36, 1;

Maple

# The function RiordanSquare is defined in A321620.
cf := proc(dim) local k, m; m := 1;
for k from dim by -1 to 1 do m := 1 - k^2*x/m od;
1/m end: RiordanSquare(cf(9), 9);

Mathematica

(* The function RiordanSquare is defined in A321620. *)
cf[dim_] := ContinuedFractionK[If[i==1, 1, -(i-1)^2 x], 1, {i, 1, dim}];
RiordanSquare[cf[9], 9] // Flatten (* Jean-François Alcover, Jul 22 2019 *)

Crossrefs

First column are the Euler numbers A000364.

Row sums are A321631, alternating row sums are A000007.

Cf. A321620.

Sequence in context: A105577 A054655 A290319 * A086745 A086646 A181612

Adjacent sequences: A321627 A321628 A321629 * A321631 A321632 A321633

Keyword

nonn,tabl

Author

Peter Luschny, Dec 09 2018

Status

approved