{"id":"n-c85abc56078671cc","revision":1,"first_index":0,"fingerprint":"f6-d583a30bb4254b7d","name":"Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","8","3"],["1","18","25","4"],["1","32","98","56","5"],["1","50","270","336","105","6"],["1","72","605","1320","891","176","7"],["1","98","1183","4004","4719","2002","273","8"],["1","128","2100","10192","18590","13728","4004","400","9"],["1","162","3468","22848","59670","68068","34476","7344","561","10"],["1","200","5415","46512","164730","268736","209950","77520","12597","760","11"],["1","242","8085","87780","405042","895356","999362","568480","159885","20482","1001","12"],["1","288","11638","155848","908523","2615008","3952228","3209328","1389223","307648","31878","1288","13"],["1","338","16250","263120","1891175","6874010","13520780","14858000","9174815","3124550","559130","47840","1625","14"],["1","392","22113","425880","3700125","16576560","41158845","58837680","49142835","23860200","6561555","968760","69615","2016","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","48","192","768","3072","12288","49152","196608","786432","3145728","12582912","50331648","201326592"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A232535","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159680,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-cff5a9f73aa6db20"]},{"id":159681,"position":1,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"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-a75bd3d9d36c6f35"]},{"id":159682,"position":2,"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-a75bd3d9d36c6f35"]},{"id":159683,"position":3,"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-a75bd3d9d36c6f35"]},{"id":159684,"position":4,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-a75bd3d9d36c6f35"]}],"region_exact":{},"incoming_edges":[{"id":"e-a75bd3d9d36c6f35","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-c85abc56078671cc","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"15c3a66935b834a44a60539a1eb95837adfea7e3af3d014bbf62bbe526b39fc4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-cff5a9f73aa6db20","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-c85abc56078671cc","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"927287ad2cd9313a43e846d4e37fdbdefe070fb7ea6aa6a8a3a5040b4f9a0fc8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[]}],"used_by":[{"id":"e-0c251aa60e82e374","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-3b09b6e351e939fe","conclusion_name":"basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-423483b3c7deec75","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-b7873c90b2011b57","conclusion_name":"basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-638c4e2ec1fd9933","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-6aadffa116a44288","conclusion_name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-a9a14b9923e7ec14","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-cca535f6115f5fc0","conclusion_name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-cb0f44f8cbe9fc7d","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-eba41671c715d7a2","conclusion_name":"basis_map_A013611 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-16137bf2f4ea840e","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-299240b586a1358b","conclusion_name":"derivative of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-3408e49d5219d076","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-54f319533dce3c44","conclusion_name":"reversal of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-a938384cba6c4260","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-cca535f6115f5fc0","conclusion_name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-ba353ec9512d926d","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-adf01e7484346696","conclusion_name":"theta_plus_one_squared of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-df9504f6b81fedcd","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-df244c0ad53b2f3d","conclusion_name":"mobius_x_over_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-e1b53362b0fdfcb4","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-0e8b59f661d3a692","conclusion_name":"theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-e2dffcc245443c80","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-6eec6efe3fe38b73","conclusion_name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-81a0a2015f78034f","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-4b5ab05dec48bd47","conclusion_name":"veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."},{"id":"e-d7bd542414b8b431","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-6b47b8135b2b438c","conclusion_name":"veronese_section_2_1 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2."}],"used_by_total":14,"links":[{"id":"l-68e1bb2e45fd41f9","kind":"fingerprint_related","node_a_id":"n-c85abc56078671cc","node_b_id":"n-54f319533dce3c44","fingerprint":"f6-d583a30bb4254b7d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d583a30bb4254b7d"},"other_id":"n-54f319533dce3c44","other":{"id":"n-54f319533dce3c44","name":"reversal of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","first_index":0,"fingerprint":"f6-d583a30bb4254b7d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["3","8","1"]],"card_rows":[["1"],["2","1"],["3","8","1"],["4","25","18","1"]],"row_sums_preview":["1","3","12","48","192","768","3072","12288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c85abc56078671cc","steps":["reverse"],"edges":["e-3408e49d5219d076"],"text":"A232535 \u2192 reverse"},"headline":"A232535 \u2192 reverse","chain":null}},{"id":"l-0de5f14629a65c8f","kind":"oeis_match","node_a_id":"n-c85abc56078671cc","node_b_id":null,"fingerprint":null,"oeis_id":"A110594","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A110594","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4de2bef938438208","kind":"oeis_match","node_a_id":"n-c85abc56078671cc","node_b_id":null,"fingerprint":null,"oeis_id":"A002001","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":"A002001","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5282388f5590232e","kind":"oeis_match","node_a_id":"n-c85abc56078671cc","node_b_id":null,"fingerprint":null,"oeis_id":"A232535","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":58},"in_catalog":true,"oeis_id":"A232535","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6c2b0b3206329cfe","kind":"oeis_match","node_a_id":"n-c85abc56078671cc","node_b_id":null,"fingerprint":null,"oeis_id":"A092898","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A092898","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a136ff5bbc63d05d","kind":"oeis_match","node_a_id":"n-c85abc56078671cc","node_b_id":null,"fingerprint":null,"oeis_id":"A164346","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":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A164346","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A232535","relation":"identification"},{"oeis_id":"A002001","relation":"match_row_sums"},{"oeis_id":"A092898","relation":"match_row_sums"},{"oeis_id":"A110594","relation":"match_row_sums"},{"oeis_id":"A164346","relation":"match_row_sums"},{"oeis_id":"A232535","relation":"match_triangle"}],"bfiles":[{"node_id":"n-c85abc56078671cc","oeis_id":"A002001","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":12947},{"node_id":"n-c85abc56078671cc","oeis_id":"A164346","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":13067},{"node_id":"n-c85abc56078671cc","oeis_id":"A232535","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":58,"oeis_terms_agreeing":58,"oeis_terms_available":58,"term_count":990,"raw_size":16748}],"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,   8,    3;\n  3: 1,  18,   25,     4;\n  4: 1,  32,   98,    56,     5;\n  5: 1,  50,  270,   336,   105,     6;\n  6: 1,  72,  605,  1320,   891,   176,     7;\n  7: 1,  98, 1183,  4004,  4719,  2002,   273,    8;\n  8: 1, 128, 2100, 10192, 18590, 13728,  4004,  400,   9;\n  9: 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*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":"A232535","target":"triangle","first_row":0,"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,   2;\n  2: 1,   8,    3;\n  3: 1,  18,   25,     4;\n  4: 1,  32,   98,    56,     5;\n  5: 1,  50,  270,   336,   105,     6;\n  6: 1,  72,  605,  1320,   891,   176,     7;\n  7: 1,  98, 1183,  4004,  4719,  2002,   273,    8;\n  8: 1, 128, 2100, 10192, 18590, 13728,  4004,  400,   9;\n  9: 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c85abc56078671cc","steps":[],"edges":[],"text":"A232535"},"headline":"A232535: Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","chain":null}