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A193591 Augmentation of the Euler partition triangle A026820.  See Comments. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 1 20:39 EST 2021. Contains 349435 sequences. (Running on oeis4.)