{"id":"n-29c226511ecc97cc","revision":0,"first_index":0,"fingerprint":"f6-5802c90b5b83bab2","name":"basis_map_A008277 of 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":[["89","140","32"],["610","1888","1040","128"],["4181","23068","23120","6592","512"],["28657","268584","438032","217600","38144","2048"],["196418","3045984","7621696","5879936","1747200","207872","8192"],["1346269","34022912","126071344","141987968","63667200","12599296","1085440","32768"],["9227465","376591124","2019540112","3196176832","2030550272","594641920","84037632","5488640","131072"],["63245986","4145389312","31673394832","68663532288","59423696384","24164933632","4982648832","528220160","27066368","524288"],["433494437","45475675252","489662939616","1428104515968","1639368595712","889553351680","251037810688","38412812288","3168337920","130809856","2097152"],["2971215073","497821016208","7495173400416","29023974029568","43358472068864","30551249944576","11335402889216","2348071452672","277189361664","18297651200","621805568","8388608"],["20365011074","5442440363328","113929222131456","580008095647488","1111701385106432","997428083427328","473351451533312","127485699588096","20199502577664","1895425638400","102413369344","2915041280","33554432"],["139583862445","59450483024696","1723212920024736","11446846731798528","27845805767969024","31346812543019008","18652271928352768","6353057220263936","1297446209519616","162283015110656","12394509107200","558312194048","13505658880","134217728"],["956722026041","649071208358108","25972066510338368","223802931488623488","685129218205375232","956749733649961984","703134328499478528","296838604723060736","75992431549743104","12166059062198272","1231571575111680","78049043283968","2976026460160","61941481472","536870912"],["6557470319842","7084159697018656","390454069568483216","4344697117593401216","16625337938910155264","28541412868827455488","25605510846155329536","13195535422084775936","4149761803122704384","826947982858387456","106535705769934848","8906229195735040","475864122785792","15558519029760","281588793344","2147483648"],["44945570212853","77302915340126476","5859175615804984496","83887258581059932480","399076431847101800704","836162749480331957248","907233353377625559040","563968944552865415168","214546859578574831616","52145644416339279872","8313738531045376000","880282463018942464","61794467509698560","2821347172417536","79973633228800","1270236577792","8589934592"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["261","3666","57473","993065","18707298","380813197","8316385069","193672652082","4785350440569","124909253223633","3431567032989634","98900292864123269","2981748060742372053","93802682407631793426","3072216483371820275697"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["89","140","32"],["610","1888","1040","128"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197551,"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-e3339e0be7ea0ef1"]},{"id":197552,"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-e3339e0be7ea0ef1"]},{"id":197553,"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-e3339e0be7ea0ef1"]}],"region_exact":{},"incoming_edges":[{"id":"e-e3339e0be7ea0ef1","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-29c226511ecc97cc","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d51ac9e0385395cb497b537116a0268d2154626388006998428b07a58f3066d0","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-b85d9f7216f04287","role":"sequence","name":"basis_map_A013611 of 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":[],"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:         89,          140,            32;\n  1:        610,         1888,          1040,            128;\n  2:       4181,        23068,         23120,           6592,            512;\n  3:      28657,       268584,        438032,         217600,          38144,           2048;\n  4:     196418,      3045984,       7621696,        5879936,        1747200,         207872,           8192;\n  5:    1346269,     34022912,     126071344,      141987968,       63667200,       12599296,        1085440,         32768;\n  6:    9227465,    376591124,    2019540112,     3196176832,     2030550272,      594641920,       84037632,       5488640,       131072;\n  7:   63245986,   4145389312,   31673394832,    68663532288,    59423696384,    24164933632,     4982648832,     528220160,     27066368,      524288;\n  8:  433494437,  45475675252,  489662939616,  1428104515968,  1639368595712,   889553351680,   251037810688,   38412812288,   3168337920,   130809856,   2097152;\n  9: 2971215073, 497821016208, 7495173400416, 29023974029568, 43358472068864, 30551249944576, 11335402889216, 2348071452672, 277189361664, 18297651200, 621805568, 8388608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 89 + 140*x + 32*x^2, P(1,x) = 610 + 1888*x + 1040*x^2 + 128*x^3.","mathematica":"P[0, x_] := 89 + 140*x + 32*x^2;\nP[1, x_] := 610 + 1888*x + 1040*x^2 + 128*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 89 + 140*x + 32*x^2; v[2] = 610 + 1888*x + 1040*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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","basis A008277"],"edges":["e-b12bbff1e34cdc3a","e-e3339e0be7ea0ef1"],"text":"A207606 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A207606 \u2192 basis A013611 \u2192 basis A008277","chain":null}