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A193590
Augmentation of the Euler triangle A008292. See Comments.
1
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.
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
Sequence in context: A336244 A083801 A344557 * A300051 A281890 A342381
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 31 2011
STATUS
approved