A193590 Augmentation of the Euler triangle A008292. See Comments.

1, 1, 1, 1, 5, 2, 1, 16, 33, 8, 1, 42, 275, 342, 58, 1, 99, 1669, 6441, 5600, 718, 1, 219, 8503, 82149, 217694, 143126, 14528, 1, 466, 39076, 843268, 5466197, 10792622, 5628738, 466220, 1, 968, 168786, 7621160, 107506633, 509354984, 788338180

Offset

0,5

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193590, (column 1)=A002662, with general term 2^n-1-n(n+1)/2.

Links

Table of n, a(n) for n=0..42.

Example

First 5 rows of A193589:
1
1....1
1....5....2
1....16...33....8
1....42...275...342....58

Mathematica

p[n_, k_] :=
Sum[((-1)^j)*((k + 1 - j)^(n + 1))*Binomial[n + 2, j], {j, 0, k + 1}]
(* A008292, Euler triangle *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193590  *)
Flatten[Table[v[n], {n, 0, 8}]]

Crossrefs

Cf. A008292, A193091.

Sequence in context: A336244 A083801 A344557 * A300051 A281890 A342381

Adjacent sequences: A193587 A193588 A193589 * A193591 A193592 A193593

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 31 2011

Status

approved