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 A193589 Augmentation of the Fibonacci triangle A193588.  See Comments. 6
 1, 1, 2, 1, 4, 7, 1, 6, 18, 31, 1, 8, 33, 90, 154, 1, 10, 52, 185, 481, 820, 1, 12, 75, 324, 1065, 2690, 4575, 1, 14, 102, 515, 2006, 6276, 15547, 26398, 1, 16, 133, 766, 3420, 12468, 37711, 92124, 156233, 1, 18, 168, 1085, 5439, 22412, 78030, 230277 (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 A193589, if the triangle is written as (w(n,k)), then w(n,n)=A007863(n); w(n,n-1)=A011270; and (col 3)=A033537. LINKS EXAMPLE First 5 rows of A193588: 1 1....2 1....2....3 1....2....3....5 1....2....3....5....8 First 5 rows of A193589: 1 1....2 1....4....7 1....6....18...31 1....8....33...90...154 MATHEMATICA p[n_, k_] := Fibonacci[k + 2] Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A193588 *) 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}]]  (* A193589 *) Flatten[Table[v[n], {n, 0, 8}]] CROSSREFS Cf. A193091, A193588. Sequence in context: A234946 A223092 A071948 * A187115 A121722 A193591 Adjacent sequences:  A193586 A193587 A193588 * A193590 A193591 A193592 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jul 31 2011 STATUS approved

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)