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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"21a6c0d503e49c5f4b3c634ad92579a6532bf96f971e0a02abba5c4d4dc1ce37","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cbce839217b23f6d","role":"sequence","name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"15"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 15]"}]},{"id":"e-4cad2095b142c753","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c72653d9ab0ee8be","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":"8c10e8f643900fba2c82854959b607434164c59bc1c59cd4263cb31cead934f7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-d339e2ece1e12101","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c72653d9ab0ee8be","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":"875b2757eb868050389e8d89b54bc3d47a201c32e3aadcf6f832cf61d102888c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-e504746edac4d52e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c72653d9ab0ee8be","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":"f551ddd62f17066395381ce6916546fde78e4ae7ea2ffb64594c31969146b846","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-32a5b77539bbfdf5","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-c72653d9ab0ee8be","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"ef53fced788cfcba2520f5597fbad4a3da33744deafae44fe24823bf42fe0222","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-13a04d0165daacbb","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-2156f740e0c97180","conclusion_name":"basis_map_A013611 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-1ebadae00ef5f3ca","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-726f071d833f296a","conclusion_name":"basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-3a20d12bade77e7f","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-c63944481deaae45","conclusion_name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-3ab95efc77e6349f","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-fa77c904daa26e1a","conclusion_name":"basis_map_A008277 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-52226b414d15e69f","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-51fcbaaf07b9691b","conclusion_name":"basis_map_A013610 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-0e7024e7b55599e9","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-2f435ee30c76fa47","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-12ad66072da87843","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-50a9a09f321fcea8","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-1365676e8cb0fd13","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-500943bf802dcc95","conclusion_name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals."},{"id":"e-25462190c1ff20a9","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-2445c52c3b45c0fd","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-2c112d9ec33e1fb9","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-305352172c67c5f0","conclusion_name":"reversal of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-4d81c1e2745a9ed0","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-383c73dff2796e24","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-4db5907b6eeb8d63","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-c63944481deaae45","conclusion_name":"basis_map_A007318 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-5274886585caff6b","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-f263ca9517b1fbce","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-61f403043a781326","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-29ef532ce8637fba","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-79b40901050ca651","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-cbce839217b23f6d","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-81aeef92852c73bb","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-d1f175c4c1bf40de","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-8a3bcadffb0d6c12","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-51f9b5eb10d6dd84","conclusion_name":"derivative of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-aa7e9b3181553556","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8d4eedf9d66a931a","conclusion_name":"theta_plus_one_squared of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-acc0c42734fdb8fc","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-95ccc13d8da74c1c","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-bf6c3a69738a7962","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-363260c438af1335","conclusion_name":"theta_plus_one of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-c9969a1cd93f190d","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f305609b43c3ec0e","conclusion_name":"one_plus_derivative of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-cc7a67409dceb768","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-cc04537b3d998791","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-f5bf3eab46ece5d6","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-35239c46f20a8933","conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-32612196073bf5c5","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-6c42f09269042b44","conclusion_name":"veronese_section_2_1 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."},{"id":"e-918cdc60ffda35ef","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-af589024565f1365","conclusion_name":"veronese_section_2_0 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."}],"used_by_total":25,"links":[{"id":"l-2b0c48c44ec88a8c","kind":"fingerprint_related","node_a_id":"n-c72653d9ab0ee8be","node_b_id":"n-305352172c67c5f0","fingerprint":"f6-4d750740fbfda5f4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4d750740fbfda5f4"},"other_id":"n-305352172c67c5f0","other":{"id":"n-305352172c67c5f0","name":"reversal of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","first_index":1,"fingerprint":"f6-4d750740fbfda5f4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","2"]],"card_rows":[["1"],["1","1"],["1","3","2"],["1","5","7","3"]],"row_sums_preview":["1","2","6","16","44","120","328","896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["reverse"],"edges":["e-2c112d9ec33e1fb9"],"text":"A063967 \u2192 reverse"},"headline":"A063967 \u2192 reverse","chain":null}},{"id":"l-06d0a7024f4f20af","kind":"oeis_match","node_a_id":"n-c72653d9ab0ee8be","node_b_id":null,"fingerprint":null,"oeis_id":"A002605","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":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002605","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-47bd690baf8d5fd7","kind":"oeis_match","node_a_id":"n-c72653d9ab0ee8be","node_b_id":null,"fingerprint":null,"oeis_id":"A063967","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A063967","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A063967","relation":"identification"},{"oeis_id":"A002605","relation":"match_row_sums"},{"oeis_id":"A063967","relation":"match_triangle"}],"bfiles":[{"node_id":"n-c72653d9ab0ee8be","oeis_id":"A063967","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":13912}],"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,   1;\n   3:  2,   3,    1;\n   4:  3,   7,    5,    1;\n   5:  5,  15,   16,    7,    1;\n   6:  8,  30,   43,   29,    9,    1;\n   7: 13,  58,  104,   95,   46,   11,   1;\n   8: 21, 109,  235,  271,  179,   67,  13,   1;\n   9: 34, 201,  506,  705,  591,  303,  92,  15,  1;\n  10: 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + x)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A063967","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  1,   1;\n  2:  2,   3,    1;\n  3:  3,   7,    5,    1;\n  4:  5,  15,   16,    7,    1;\n  5:  8,  30,   43,   29,    9,    1;\n  6: 13,  58,  104,   95,   46,   11,   1;\n  7: 21, 109,  235,  271,  179,   67,  13,   1;\n  8: 34, 201,  506,  705,  591,  303,  92,  15,  1;\n  9: 55, 365, 1051, 1717, 1746, 1140, 475, 121, 17, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + x)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (1 + x)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":[],"edges":[],"text":"A063967"},"headline":"A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","chain":null}