{"id":"n-1fe6eede339bc6ea","revision":1,"first_index":0,"fingerprint":"f6-776041ebb0e099b1","name":"Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3"],["8","1"],["21","6"],["55","25","1"],["144","90","9"],["377","300","51","1"],["987","954","234","12"],["2584","2939","951","86","1"],["6765","8850","3573","480","15"],["17711","26195","12707","2305","130","1"],["46368","76500","43398","10008","855","18"],["121393","221016","143682","40426","4740","183","1"],["317811","632916","464148","154668","23373","1386","21"],["832040","1799125","1469778","567260","105805","8715","245","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","9","27","81","243","729","2187","6561","19683","59049","177147","531441","1594323","4782969"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A181371","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160387,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]","region_exact":null,"edges":["e-6b3e76cd98792eb8"]}],"region_exact":{},"incoming_edges":[{"id":"e-6b3e76cd98792eb8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1/9\",\"intercept\":\"-1/9\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-1fe6eede339bc6ea","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"dab5108632dba10be77537fb8390ffcdafc4186a71d4d6c669b9f7b19de7bb79","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}}],"premises":[{"position":0,"node_id":"n-75ccd0453a9b6cec","role":"sequence","name":"Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-24bbc67eaf9c9c2b","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A133494","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A133494","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2cc45b82230c6ea8","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A354269","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A354269","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-617124f19f17c7b2","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A027313","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A027313","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-63ac3ca9df2f96ae","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A155734","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A155734","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c2b996814d6fce64","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A376478","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A376478","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-caa61be70109d849","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A140429","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A140429","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e16ed99d7b976a42","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A181371","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":13,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":13,"target":"triangle","terms_compared":50},"in_catalog":true,"oeis_id":"A181371","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-eb151c710e3fcd40","kind":"oeis_match","node_a_id":"n-1fe6eede339bc6ea","node_b_id":null,"fingerprint":null,"oeis_id":"A000244","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000244","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A181371","relation":"identification"},{"oeis_id":"A000244","relation":"match_row_sums"},{"oeis_id":"A027313","relation":"match_row_sums"},{"oeis_id":"A133494","relation":"match_row_sums"},{"oeis_id":"A140429","relation":"match_row_sums"},{"oeis_id":"A155734","relation":"match_row_sums"},{"oeis_id":"A354269","relation":"match_row_sums"},{"oeis_id":"A376478","relation":"match_row_sums"},{"oeis_id":"A181371","relation":"match_triangle"}],"bfiles":[{"node_id":"n-1fe6eede339bc6ea","oeis_id":"A000244","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":28,"oeis_terms_available":28,"term_count":200,"raw_size":10484},{"node_id":"n-1fe6eede339bc6ea","oeis_id":"A181371","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":50,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":992,"raw_size":19355}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    3;\n  2:    8,    1;\n  3:   21,    6;\n  4:   55,   25,    1;\n  5:  144,   90,    9;\n  6:  377,  300,   51,   1;\n  7:  987,  954,  234,  12;\n  8: 2584, 2939,  951,  86,  1;\n  9: 6765, 8850, 3573, 480, 15;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 3*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = 3*v[i - 1] - (1 - x)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A181371","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    3;\n  2:    8,    1;\n  3:   21,    6;\n  4:   55,   25,    1;\n  5:  144,   90,    9;\n  6:  377,  300,   51,   1;\n  7:  987,  954,  234,  12;\n  8: 2584, 2939,  951,  86,  1;\n  9: 6765, 8850, 3573, 480, 15;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 3*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = 3*v[i - 1] - (1 - x)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["x \u21a6 (1/9)x\u22121/9"],"edges":["e-6b3e76cd98792eb8"],"text":"A011973 \u2192 x \u21a6 (1/9)x\u22121/9"},"headline":"A181371: Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).","chain":"A011973 \u2192 x \u21a6 (1/9)x\u22121/9"}
