{"id":"n-e189038e95d51606","revision":2,"first_index":0,"fingerprint":"f6-3a3e895c37a3d3df","name":"Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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"],["1"],["1","3"],["1","6"],["1","9","9"],["1","12","27"],["1","15","54","27"],["1","18","90","108"],["1","21","135","270","81"],["1","24","189","540","405"],["1","27","252","945","1215","243"],["1","30","324","1512","2835","1458"],["1","33","405","2268","5670","5103","729"],["1","36","495","3240","10206","13608","5103"],["1","39","594","4455","17010","30618","20412","2187"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","7","19","40","97","217","508","1159","2683","6160","14209","32689","75316"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A304236","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":"e-46ee23218258c2c9","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159895,"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-0050d566ad712b7d","e-64b74724257dbeaa"]},{"id":159896,"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-0050d566ad712b7d","e-64b74724257dbeaa"]},{"id":159897,"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-0050d566ad712b7d","e-64b74724257dbeaa"]},{"id":159898,"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-0050d566ad712b7d"]},{"id":159899,"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, 0]","statement":"every PF input has a real-rooted image with every root <= -c/b = 0"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: every PF input has a real-rooted image with every root <= -c/b = 0","region_exact":null,"edges":["e-928936169e50a847"]},{"id":159900,"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, 0]","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, 0]: every PF input has a PF image (zero allowed)","region_exact":null,"edges":["e-928936169e50a847"]}],"region_exact":{},"incoming_edges":[{"id":"e-928936169e50a847","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-e189038e95d51606","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":"7f03b85b72fca337649e33901479c7c042a6f78f28803ce2963955b185ed2a54","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"every PF input has a real-rooted image with every root <= -c/b = 0"},{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"every PF input has a PF image (zero allowed)"}],"premises":[]},{"id":"e-46ee23218258c2c9","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-e189038e95d51606","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"859516f42b4a18617f2b11e387a7357c1b38be1cd5da664b956cf38a307b0269","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-64b74724257dbeaa","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-e189038e95d51606","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"bf125c02e7e084a241e89da4187b69bc01508ff54a39c2c1956befd03dc1cd89","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-d8f9ad9ce0e039d0","role":"sequence","name":"Pascal-(1,3,1) array.","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]"}]},{"id":"e-0050d566ad712b7d","operation":{"operation":"parent_provers","route":"paper_theorem"},"operation_detail":"paper_theorem","conclusion_id":"n-e189038e95d51606","premise_count":0,"theorem":"polytool parent provers, route paper_theorem: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9512eddffa6eb79b68454c8f9d408750984a9487d8198b7dd9368d8d151b56c8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-3091186ded9433db","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-4d868f67dfcfb6b7","conclusion_name":"basis_map_A007318 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-7482884421cc3299","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-77dcaf943408789b","conclusion_name":"basis_map_A013611 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-ab47cf01b7fbfda0","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-8d5fed5e270db201","conclusion_name":"basis_map_A013610 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-c25a9d9d3ec02f57","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-7ade5c0c37e8b043","conclusion_name":"basis_map_A008277 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-c77d0879f7f3e65b","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-f074ae9ffad82cf4","conclusion_name":"basis_map_A013609 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-4cdd530d16220003","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-4d868f67dfcfb6b7","conclusion_name":"basis_map_A007318 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-4e4cfdea6d7e0f6f","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-24dd20155389652b","conclusion_name":"theta_plus_one_squared of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-825bd9c47e6b8dee","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2acd092e2d70afe2","conclusion_name":"theta_plus_one of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-8635ad6e77f468ea","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-3cd70e8420d70047","conclusion_name":"one_plus_derivative of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-916dd5eda11218db","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-111aaf6bff5783e2","conclusion_name":"derivative of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-1c5b9a0a4dd25652","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-21c6f4e12b73105f","conclusion_name":"veronese_section_2_0 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."},{"id":"e-5a301867196b7d95","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-d852a326c73cb994","conclusion_name":"veronese_section_2_1 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows."}],"used_by_total":12,"links":[{"id":"l-5a3b87453b8ceb38","kind":"oeis_match","node_a_id":"n-e189038e95d51606","node_b_id":null,"fingerprint":null,"oeis_id":"A006130","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":"A006130","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a3805f5c77685c0e","kind":"oeis_match","node_a_id":"n-e189038e95d51606","node_b_id":null,"fingerprint":null,"oeis_id":"A304236","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":15,"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":15,"target":"triangle","terms_compared":64},"in_catalog":true,"oeis_id":"A304236","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A304236","relation":"identification"},{"oeis_id":"A006130","relation":"match_row_sums"},{"oeis_id":"A304236","relation":"match_triangle"}],"bfiles":[{"node_id":"n-e189038e95d51606","oeis_id":"A006130","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":33,"oeis_terms_available":33,"term_count":200,"raw_size":8160},{"node_id":"n-e189038e95d51606","oeis_id":"A304236","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":64,"oeis_terms_agreeing":64,"oeis_terms_available":64,"term_count":992,"raw_size":15280}],"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;\n  2: 1,  3;\n  3: 1,  6;\n  4: 1,  9,   9;\n  5: 1, 12,  27;\n  6: 1, 15,  54,  27;\n  7: 1, 18,  90, 108;\n  8: 1, 21, 135, 270,  81;\n  9: 1, 24, 189, 540, 405;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + 3*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 3*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + 3*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":"A304236","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: 1;\n  2: 1,  3;\n  3: 1,  6;\n  4: 1,  9,   9;\n  5: 1, 12,  27;\n  6: 1, 15,  54,  27;\n  7: 1, 18,  90, 108;\n  8: 1, 21, 135, 270,  81;\n  9: 1, 24, 189, 540, 405;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + 3*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 3*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + 3*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-e189038e95d51606","steps":[],"edges":[],"text":"A304236"},"headline":"A304236: Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","chain":null}