{"id":"n-250c2dfa188418f4","revision":0,"first_index":0,"fingerprint":"f6-fef279a35ea92077","name":"basis_map_A013611 of A127899(unsigned) * A007318.","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":true,"rows":[["1"],["6","8"],["18","60","48"],["48","256","448","256"],["120","880","2400","2880","1280"],["288","2688","9984","18432","16896","6144"],["672","7616","35840","89600","125440","93184","28672"],["1536","20480","116736","368640","696320","786432","491520","131072"],["3456","52992","354816","1354752","3225600","4902912","4644864","2506752","589824"],["7680","133120","1024000","4587520","13189120","25231360","32112640","26214400","12451840","2621440"],["16896","326656","2838528","14598144","49201152","113541120","181665792","198967296","142737408","60555264","11534336"],["36864","786432","7618560","44236800","171048960","462422016","891813888","1226833920","1179648000","754974720","289406976","50331648"],["79872","1863680","19914752","128860160","562298880","1743126528","3936092160","6522667008","7872184320","6747586560","3898605568","1363148800","218103808"],["172032","4358144","50921472","363331584","1766195200","6176636928","15986589696","31004295168","45053116416","48444211200","37463523328","19730006016","6341787648","939524096"],["368640","10076160","127795200","996802560","5341839360","20805058560","60728279040","134951731200","229417943040","296893808640","287897026560","202836541440","98146713600","29192355840","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","126","1008","7560","54432","381024","2612736","17635968","117573120","775982592","5079158784","33014532096","213324668928","1371372871680"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","8"],["18","60","48"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","32","32"],["-4","-16","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-16"],["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204596,"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-fae6d4b95ca59aad"]},{"id":204597,"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-fae6d4b95ca59aad"]},{"id":204598,"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-fae6d4b95ca59aad"]},{"id":204599,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-fae6d4b95ca59aad"]}],"region_exact":{},"incoming_edges":[{"id":"e-fae6d4b95ca59aad","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-250c2dfa188418f4","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"84906e7955584e2f5694a66542d9a494b4973e32f04eedaeb7f75b2597ad0bf9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e917c954825de4ec","role":"sequence","name":"A127899(unsigned) * A007318.","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-10f48b5d32cbd4fa","kind":"fingerprint_related","node_a_id":"n-5484fa124567214f","node_b_id":"n-250c2dfa188418f4","fingerprint":"f6-fef279a35ea92077","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fef279a35ea92077"},"other_id":"n-5484fa124567214f","other":{"id":"n-5484fa124567214f","name":"basis_map_A007318 of Pascal (1,3) triangle.","first_index":0,"fingerprint":"f6-fef279a35ea92077","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["4","3"],["8","10","3"]],"card_rows":[["3"],["4","3"],["8","10","3"],["16","28","16","3"]],"row_sums_preview":["3","7","21","63","189","567","1701","5103"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c5cd4d1a5bdd7e74","steps":["basis A007318"],"edges":["e-2af864e906e9c0e4"],"text":"A095660 \u2192 basis A007318"},"headline":"A095660 \u2192 basis A007318","chain":null}},{"id":"l-13b505a945608b14","kind":"fingerprint_related","node_a_id":"n-3f5c139f0b8b7b9d","node_b_id":"n-250c2dfa188418f4","fingerprint":"f6-fef279a35ea92077","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fef279a35ea92077"},"other_id":"n-3f5c139f0b8b7b9d","other":{"id":"n-3f5c139f0b8b7b9d","name":"basis_map_A013611 of Numbers in (2,1)-Pascal triangle (by row).","first_index":0,"fingerprint":"f6-fef279a35ea92077","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","4"],["6","20","16"]],"card_rows":[["1"],["3","4"],["6","20","16"],["12","64","112","64"]],"row_sums_preview":["1","7","42","252","1512","9072","54432","326592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c5f5be79d9d1ae3","steps":["basis A013611"],"edges":["e-d5954e47e6d7a9a4"],"text":"A029653 \u2192 basis A013611"},"headline":"A029653 \u2192 basis A013611","chain":null}}],"oeis":[],"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:    6,      8;\n  2:   18,     60,      48;\n  3:   48,    256,     448,     256;\n  4:  120,    880,    2400,    2880,     1280;\n  5:  288,   2688,    9984,   18432,    16896,     6144;\n  6:  672,   7616,   35840,   89600,   125440,    93184,    28672;\n  7: 1536,  20480,  116736,  368640,   696320,   786432,   491520,   131072;\n  8: 3456,  52992,  354816, 1354752,  3225600,  4902912,  4644864,  2506752,   589824;\n  9: 7680, 133120, 1024000, 4587520, 13189120, 25231360, 32112640, 26214400, 12451840, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(16*n-32)*x+(16*n-32)*x^2)*P(n-2,x) + (4*n-8+(8*n-16)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 8*x, P(2,x) = 18 + 60*x + 48*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 8*x;\nP[2, x_] := 18 + 60*x + 48*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (16*n - 32)*x + (16*n - 32)*x^2)*P[n - 2, x] + (4*n - 8 + (8*n - 16)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 8*x; v[3] = 18 + 60*x + 48*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (16*n - 32)*x + (16*n - 32)*x^2)*v[i - 2] + (4*n - 8 + (8*n - 16)*x)*v[i - 1])/(n - 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-e917c954825de4ec","steps":["basis A013611"],"edges":["e-fae6d4b95ca59aad"],"text":"A134239 \u2192 basis A013611"},"headline":"A134239 \u2192 basis A013611","chain":null}
