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Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","6"],["4","24","36"]],"card_rows":[["1"],["2","6"],["4","24","36"],["8","72","216","216"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013610"],"edges":["e-db1885975295b772"],"text":"A130123 \u2192 basis A013610"},"headline":"A130123 \u2192 basis A013610","chain":null}},{"id":"l-f9eb9128713d7343","kind":"fingerprint_related","node_a_id":"n-4575a0f9ca60b9ab","node_b_id":"n-1cfea80692e421ba","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-4575a0f9ca60b9ab","other":{"id":"n-4575a0f9ca60b9ab","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","9"],["3","18","27"]],"card_rows":[["3"],["3","9"],["3","18","27"],["3","27","81","81"]],"row_sums_preview":["3","12","48","192","768","3072","12288","49152"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["x \u21a6 x/(x+1)"],"edges":["e-e61ce0e3d93770a8"],"text":"A120909 \u2192 x \u21a6 x/(x+1)"},"headline":"A120909 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f28ebd7c29653597","kind":"oeis_match","node_a_id":"n-1cfea80692e421ba","node_b_id":null,"fingerprint":null,"oeis_id":"A262710","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A262710","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A262710","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-1cfea80692e421ba","oeis_id":"A262710","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":13072}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2: -1,  -3;\n   3:  1,   6,    9;\n   4: -1,  -9,  -27,   -27;\n   5:  1,  12,   54,   108,     81;\n   6: -1, -15,  -90,  -270,   -405,   -243;\n   7:  1,  18,  135,   540,   1215,   1458,    729;\n   8: -1, -21, -189,  -945,  -2835,  -5103,  -5103,  -2187;\n   9:  1,  24,  252,  1512,   5670,  13608,  20412,  17496,   6561;\n  10: -1, -27, -324, -2268, -10206, -30618, -61236, -78732, -59049, -19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+3*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(1 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(1 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013610"],"edges":["e-fe4cc1f9dd1d3096"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013610","chain":null}