{"id":"n-b85d9f7216f04287","revision":1,"first_index":0,"fingerprint":"f6-d4f5100b4a0a6575","name":"basis_map_A013611 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["13","44","32"],["34","176","272","128"],["89","620","1488","1472","512"],["233","2040","6672","10240","7424","2048"],["610","6432","26688","55936","62720","35840","8192"],["1597","19696","99120","264320","404480","356352","167936","32768"],["4181","59044","349456","1133504","2208000","2651136","1921024","770048","131072"],["10946","174160","1185424","4534016","10753536","16429056","16199680","9961472","3473408","524288"],["28657","507220","3903456","17210240","48188672","89650176","112394240","93913088","50135040","15466496","2097152"],["75025","1462128","12553824","62710528","202653440","445276160","679583744","721354752","522584064","246415360","68157440","8388608"],["196418","4179264","39606528","221136640","810613760","2056792064","3707461632","4788486144","4403036160","2814115840","1188036608","297795584","33554432"],["514229","11861336","122982816","759125504","3113734400","8967555072","18669969408","28473950208","31840468992","25808076800","14752415744","5637144576","1291845632","134217728"],["1346269","33461660","376787264","2548171136","11567091456","37300810752","88172666880","155313242112","205014171648","201971990528","146301517824","75623301120","26390560768","5570035712","536870912"],["3524578","93908720","1141225616","8392536960","41780224512","149203243008","395051274240","790156443648","1204455014400","1400164581376","1232040099840","806438830080","380373041152","122138132480","23890755584","2147483648"],["9227465","262362812","3422524464","27194342208","147343729920","577429816320","1693794127872","3795361185792","6568976646144","8816341811200","9150477107200","7271853588480","4340483883008","1882336526336","559687925760","102005473280","8589934592"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["89","610","4181","28657","196418","1346269","9227465","63245986","433494437","2971215073","20365011074","139583862445","956722026041","6557470319842","44945570212853"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["13","44","32"],["34","176","272","128"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167879,"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-b12bbff1e34cdc3a"]},{"id":167880,"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-b12bbff1e34cdc3a"]},{"id":167881,"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-b12bbff1e34cdc3a"]},{"id":167882,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-b12bbff1e34cdc3a"]}],"region_exact":{},"incoming_edges":[{"id":"e-b12bbff1e34cdc3a","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-b85d9f7216f04287","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":"b4bb7e6c7709ce41f06556f7f3999c6033208824f135b79b8c5cf2f669673af7","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":"unordered"}],"premises":[{"position":0,"node_id":"n-dfab35176bd08e72","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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-c65b837e36f2ac04","kind":"oeis_match","node_a_id":"n-b85d9f7216f04287","node_b_id":null,"fingerprint":null,"oeis_id":"A033891","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":"A033891","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-cfc3f34dd5a2d21a","kind":"oeis_match","node_a_id":"n-b85d9f7216f04287","node_b_id":null,"fingerprint":null,"oeis_id":"A172968","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A172968","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A033891","relation":"match_row_sums"},{"oeis_id":"A172968","relation":"match_row_sums"}],"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:    13,      44,       32;\n  1:    34,     176,      272,      128;\n  2:    89,     620,     1488,     1472,       512;\n  3:   233,    2040,     6672,    10240,      7424,      2048;\n  4:   610,    6432,    26688,    55936,     62720,     35840,      8192;\n  5:  1597,   19696,    99120,   264320,    404480,    356352,    167936,     32768;\n  6:  4181,   59044,   349456,  1133504,   2208000,   2651136,   1921024,    770048,    131072;\n  7: 10946,  174160,  1185424,  4534016,  10753536,  16429056,  16199680,   9961472,   3473408,    524288;\n  8: 28657,  507220,  3903456, 17210240,  48188672,  89650176, 112394240,  93913088,  50135040,  15466496,  2097152;\n  9: 75025, 1462128, 12553824, 62710528, 202653440, 445276160, 679583744, 721354752, 522584064, 246415360, 68157440, 8388608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 44*x + 32*x^2, P(1,x) = 34 + 176*x + 272*x^2 + 128*x^3.","mathematica":"P[0, x_] := 13 + 44*x + 32*x^2;\nP[1, x_] := 34 + 176*x + 272*x^2 + 128*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 44*x + 32*x^2; v[2] = 34 + 176*x + 272*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + 4*x)*v[i - 1]); 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-dfab35176bd08e72","steps":["basis A013611"],"edges":["e-b12bbff1e34cdc3a"],"text":"A207606 \u2192 basis A013611"},"headline":"A207606 \u2192 basis A013611","chain":null}