{"id":"n-88b9d8361c4b11db","revision":1,"first_index":0,"fingerprint":"f6-d5b12dcc66b60731","name":"reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","4","1"],["1","6","3"],["1","8","6"],["1","10","10"],["1","12","15"],["1","14","21"],["1","16","28"],["1","18","36"],["1","20","45"],["1","22","55"],["1","24","66"],["1","26","78"],["1","28","91"]],"degrees":[0,1,2,2,2,2,2,2,2,2,2,2,2,2,2],"row_sums":["1","3","6","10","15","21","28","36","45","55","66","78","91","105","120"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"],["1","4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160999,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-29732eeeac05382b"]},{"id":161000,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-29732eeeac05382b"]},{"id":161001,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-29732eeeac05382b"]}],"region_exact":{},"incoming_edges":[{"id":"e-29732eeeac05382b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-88b9d8361c4b11db","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"1dca936b1d223fb542329aea859835a282cea22f8c9b2f87c0b24447dc7cfbbb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1815d3e1c9c57d5b","role":"sequence","name":"Number triangle T(n,k)=C(n,k)C(2,n-k).","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-2d3c9c8007bd92cd","kind":"fingerprint_related","node_a_id":"n-fdf7b22232b1fd38","node_b_id":"n-88b9d8361c4b11db","fingerprint":"f6-d5b12dcc66b60731","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d5b12dcc66b60731"},"other_id":"n-fdf7b22232b1fd38","other":{"id":"n-fdf7b22232b1fd38","name":"derivative of Number triangle T(n,k)=C(n,k)C(2,n-k).","first_index":0,"fingerprint":"f6-d5b12dcc66b60731","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["3","12","3"],["0","12","24","4"]],"row_sums_preview":["0","1","6","18","40","75","126","196"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["(1/2)D\u00b2+2D+1","D"],"edges":["e-0c9044372fd705ed","e-2c8197af5d9a4024"],"text":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 D"},"headline":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 D","chain":null}},{"id":"l-33b30d168c9c8af1","kind":"fingerprint_related","node_a_id":"n-1815d3e1c9c57d5b","node_b_id":"n-88b9d8361c4b11db","fingerprint":"f6-d5b12dcc66b60731","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d5b12dcc66b60731"},"other_id":"n-1815d3e1c9c57d5b","other":{"id":"n-1815d3e1c9c57d5b","name":"Number triangle T(n,k)=C(n,k)C(2,n-k).","first_index":0,"fingerprint":"f6-d5b12dcc66b60731","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["1","4","1"]],"card_rows":[["1"],["2","1"],["1","4","1"],["0","3","6","1"]],"row_sums_preview":["1","3","6","10","15","21","28","36"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A106246"],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["(1/2)D\u00b2+2D+1"],"edges":["e-0c9044372fd705ed"],"text":"A023531 \u2192 (1/2)D\u00b2+2D+1"},"headline":"A106246: Number triangle T(n,k)=C(n,k)C(2,n-k).","chain":"A023531 \u2192 (1/2)D\u00b2+2D+1"}},{"id":"l-2ea37516414dbbc4","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A000217","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":"A000217","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-346fc5d6d6e6e46f","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A105337","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":"A105337","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-75779ed61421d036","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A105336","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":"A105336","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-81d5ddc30e564db2","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A025738","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":"A025738","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a98aac719b500264","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A025747","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":"A025747","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bacbc08e8d2567d1","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A078256","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A078256","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d4ae4d5e466b3fef","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A025731","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":"A025731","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d7b6fd4ce9d55865","kind":"oeis_match","node_a_id":"n-88b9d8361c4b11db","node_b_id":null,"fingerprint":null,"oeis_id":"A077367","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A077367","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A000217","relation":"match_row_sums"},{"oeis_id":"A025731","relation":"match_row_sums"},{"oeis_id":"A025738","relation":"match_row_sums"},{"oeis_id":"A025747","relation":"match_row_sums"},{"oeis_id":"A077367","relation":"match_row_sums"},{"oeis_id":"A078256","relation":"match_row_sums"},{"oeis_id":"A105336","relation":"match_row_sums"},{"oeis_id":"A105337","relation":"match_row_sums"}],"bfiles":[],"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: 1,  2;\n  2: 1,  4,  1;\n  3: 1,  6,  3;\n  4: 1,  8,  6;\n  5: 1, 10, 10;\n  6: 1, 12, 15;\n  7: 1, 14, 21;\n  8: 1, 16, 28;\n  9: 1, 18, 36;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-3,x) - 3*P(n-2,x) + 3*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[P[n - 3, x] - 3*P[n - 2, x] + 3*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = v[i - 3] - 3*v[i - 2] + 3*v[i - 1]); 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-6fea5aff4f44fa54","steps":["(1/2)D\u00b2+2D+1","reverse"],"edges":["e-0c9044372fd705ed","e-29732eeeac05382b"],"text":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 reverse"},"headline":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 reverse","chain":null}
