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A193606 Augmentation of the triangle A193605.  See Comments. 2
1, 1, 3, 1, 7, 17, 1, 12, 57, 127, 1, 18, 134, 531, 1125, 1, 25, 265, 1556, 5513, 11279, 1, 33, 470, 3793, 19152, 62675, 124837, 1, 42, 772, 8175, 55297, 250524, 771121, 1502679, 1, 52, 1197, 16087, 140269, 834879, 3478204, 10185019, 19480445 (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.

LINKS

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

EXAMPLE

First five rows of A193606:

1

1...3

1...7....17

1...12...57....127

1...18...134...531...1125

MATHEMATICA

u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]

p[n_, k_] := Sum[u[n, h], {h, 0, k}]

Table[p[n, k], {n, 0, 12}, {k, 0, n}] (* A193695 *)

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}]] (* A193606 *)

Flatten[Table[v[n], {n, 0, 8}]]

CROSSREFS

Cf. A193605, A193091.

Sequence in context: A232295 A228763 A101845 * A221328 A280610 A280416

Adjacent sequences:  A193603 A193604 A193605 * A193607 A193608 A193609

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jul 31 2011

STATUS

approved

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Last modified June 4 01:54 EDT 2020. Contains 334812 sequences. (Running on oeis4.)