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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 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.)