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 A144401 Padovan ( A000931) version of A038137: expansion of polynomials as antidiagonal: p(x,n)=1/(1-x-x^3)^n. 0
 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 6, 3, 1, 5, 10, 13, 11, 4, 1, 6, 15, 24, 27, 18, 6, 1, 7, 21, 40, 55, 51, 30, 9, 1, 8, 28, 62, 100, 116, 94, 50, 13, 1, 9, 36, 91, 168, 231, 234, 171, 81, 19, 1, 10, 45, 128, 266, 420, 505, 460, 303, 130, 28, 1, 11, 55, 174, 402, 714, 987, 1065 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332 (cf. A008998). These polynomials are sort of pseudo-combinations with the last element Padovan instead of one. If you subtract the binomial triangle sequence you get: {0}, {0, 0}, {0, 0, 0}, {0, 0, 0, 1}, {0, 0, 0, 2, 2}, {0, 0, 0, 3, 6, 3}, {0, 0, 0, 4, 12, 12, 5}, {0, 0, 0, 5, 20, 30, 23, 8}, {0, 0, 0, 6, 30, 60, 66, 42, 12} LINKS FORMULA p(x,n)=1/(1-x-x^3)^n; t(n,m)=anti_diagonal_expansion(p(x,n)). EXAMPLE {1}, {1, 1}, {1, 2, 1}, {1, 3, 3, 2}, {1, 4, 6, 6, 3}, {1, 5, 10, 13, 11, 4}, {1, 6, 15, 24, 27, 18, 6}, {1, 7, 21, 40, 55, 51, 30, 9}, {1, 8, 28, 62, 100, 116, 94, 50, 13}, {1, 9, 36, 91, 168, 231, 234, 171, 81, 19}, {1, 10, 45, 128, 266, 420, 505, 460, 303, 130, 28}, {1, 11, 55, 174, 402, 714, 987, 1065, 879, 527, 208, 41}, {1, 12, 66, 230, 585, 1152, 1792, 2220, 2175, 1640, 906, 330, 60}, {1, 13, 78, 297, 825, 1782, 3072, 4278, 4815, 4320, 3006, 1539, 520, 88}, {1, 14, 91, 376, 1133, 2662, 5028, 7752, 9807, 10122, 8391, 5424, 2586, 816, 129} MATHEMATICA Clear[f, b, a, g, h, n, t]; f[t_, n_] = 1/(1 - t - t^3)^n; a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]; Flatten[b] CROSSREFS Cf. A000931, A038137. Sequence in context: A242779 A215065 A175424 * A034929 A099509 A153859 Adjacent sequences:  A144398 A144399 A144400 * A144402 A144403 A144404 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula and Gary W. Adamson, Oct 03 2008 STATUS approved

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Last modified July 24 12:47 EDT 2021. Contains 346273 sequences. (Running on oeis4.)