{"id":"n-72cbbb2f152c8f9c","revision":0,"first_index":0,"fingerprint":"f6-4899378506777ad9","name":"Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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"],["2"],["3","1"],["5","3"],["8","7","1"],["13","15","4"],["21","30","12","1"],["34","58","31","5"],["55","109","73","18","1"],["89","201","162","54","6"],["144","365","344","145","25","1"],["233","655","707","361","85","7"],["377","1164","1416","850","255","33","1"],["610","2052","2778","1918","701","125","8"],["987","3593","5358","4184","1806","413","42","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","4","8","16","32","64","128","256","512","1024","2048","4096","8192","16384"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A209599","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169725,"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-e6a18deaef81a66e"]},{"id":169726,"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-e6a18deaef81a66e"]},{"id":169727,"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-e6a18deaef81a66e"]},{"id":169728,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-e6a18deaef81a66e"]},{"id":169729,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, -1]","statement":"every PF input has a real-rooted image with every root <= -c/b = -1"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, -1]: every PF input has a real-rooted image with every root <= -c/b = -1","region_exact":null,"edges":["e-a5c879407de7a175"]},{"id":169730,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, -1]","statement":"every PF input has a PF image (zero allowed)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, -1]: every PF input has a PF image (zero allowed)","region_exact":null,"edges":["e-a5c879407de7a175"]}],"region_exact":{},"incoming_edges":[{"id":"e-a5c879407de7a175","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-72cbbb2f152c8f9c","premise_count":0,"theorem":"constant current multiplier: P_0 = 1, P_1 = s, P_(k+2) = a P_(k+1) + (b t + c) P_k with a, s >= 0, b > 0.  P_k(t) = Q_k(b t + c) for the rows Q_k with b = 1, c = 0, whose map sends PF inputs to PF outputs (an interlacing-preserving 2 x 2 matrix, Branden, arXiv:1410.6601, Theorem 8.5), so every PF input has a real-rooted image with roots <= -c/b; PF inputs (also those with roots in any [-R, 0]) have PF images if and only if every P_k(0) >= 0, if and only if a^2 + 4c >= 0 and (2s >= a or c >= s^2 - a s) (RealRooted orderTwoBasis_pf_equivalences, basisTransform_orderTwoBasis_preserves_bounded_pf_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"a9026baa767e8e596fd760588c0511344c06442c2db6d4da63cbef9f4ebfe88d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, -1]","statement":"every PF input has a real-rooted image with every root <= -c/b = -1"},{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, -1]","statement":"every PF input has a PF image (zero allowed)"}],"premises":[]},{"id":"e-e6a18deaef81a66e","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-72cbbb2f152c8f9c","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":null,"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"certificate_hash":"69b9129f3cd354d1ed9fbe42485ebfa9e0ff3a387ff670a30f7875c9373daf74","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-3ef3c280391234a6","kind":"fingerprint_related","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":"n-540fd96c9759fa70","fingerprint":"f6-4899378506777ad9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4899378506777ad9"},"other_id":"n-540fd96c9759fa70","other":{"id":"n-540fd96c9759fa70","name":"A skewed version of triangular array A122075.","first_index":0,"fingerprint":"f6-4899378506777ad9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","1","3"]],"card_rows":[["1"],["0","2"],["0","1","3"],["0","0","3","5"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":["A236076"],"derivation":{"root":"n-540fd96c9759fa70","steps":[],"edges":[],"text":"A236076"},"headline":"A236076: A skewed version of triangular array A122075.","chain":null}},{"id":"l-0411f1dbc9714f2f","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A166444","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A166444","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5b134013388478af","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A118655","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":"A118655","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6eb2e4b078495b00","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A220051","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":"A220051","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-98effea0c1526c23","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A034008","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034008","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c013d1e5a6f5260e","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A011782","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":"A011782","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c23d49ea4fce2304","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A050732","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":"A050732","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e1ea9012874a7c8e","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A000079","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":"A000079","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fb3efe6475ea07e5","kind":"oeis_match","node_a_id":"n-72cbbb2f152c8f9c","node_b_id":null,"fingerprint":null,"oeis_id":"A131577","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":"A131577","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A209599","relation":"identification"},{"oeis_id":"A000079","relation":"match_row_sums"},{"oeis_id":"A011782","relation":"match_row_sums"},{"oeis_id":"A034008","relation":"match_row_sums"},{"oeis_id":"A050732","relation":"match_row_sums"},{"oeis_id":"A118655","relation":"match_row_sums"},{"oeis_id":"A131577","relation":"match_row_sums"},{"oeis_id":"A166444","relation":"match_row_sums"},{"oeis_id":"A220051","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-72cbbb2f152c8f9c","oeis_id":"A000079","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":35,"oeis_terms_available":35,"term_count":200,"raw_size":6981}],"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:  2;\n  2:  3,   1;\n  3:  5,   3;\n  4:  8,   7,   1;\n  5: 13,  15,   4;\n  6: 21,  30,  12,  1;\n  7: 34,  58,  31,  5;\n  8: 55, 109,  73, 18, 1;\n  9: 89, 201, 162, 54, 6;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[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] = 2; for(n = 2, N, my(i = n + 1); v[i] = 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-72cbbb2f152c8f9c","steps":[],"edges":[],"text":"A209599"},"headline":"A209599: Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","chain":null}
