A193591 Augmentation of the Euler partition triangle A026820. See Comments.

1, 1, 2, 1, 4, 7, 1, 7, 19, 31, 1, 10, 45, 103, 161, 1, 14, 82, 297, 617, 937, 1, 18, 146, 652, 2057, 4005, 5953, 1, 23, 228, 1395, 5251, 15004, 27836, 40668, 1, 28, 355, 2555, 13023, 43470, 115110, 205516, 295922, 1, 34, 509, 4689, 27327, 122006, 371942

Offset

0,3

Comments

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

Regarding A193591, column 1)=A014616.

Links

Table of n, a(n) for n=0..51.

Example

First 5 rows of A193589:
1
1...2
1...4...7
1...7...19...31
1...10..45...103...161

Mathematica

p[n_, k_] := Length@IntegerPartitions[n + 1,
   k + 1] (* A026820, Euler partition 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, 12}]]  (* A193591 *)
Flatten[Table[v[n], {n, 0, 9}]]

Crossrefs

Cf. A026820, A193091.

Sequence in context: A193589 A187115 A121722 * A218842 A219421 A297314

Adjacent sequences: A193588 A193589 A193590 * A193592 A193593 A193594

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 31 2011

Status

approved