{"id":"n-43f67bf0103635e1","revision":0,"first_index":0,"fingerprint":"f6-0f7f9997c3a05a90","name":"shift_x_plus_one of basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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"],["3","1"],["27","21","4"],["513","642","265","36"],["16929","29394","19017","5428","576"],["863379","1922319","1704717","752253","165076","14400"],["63026667","171410931","193647825","116284281","39131656","6993936","518400"],["6239640033","20057988852","27570270294","21000887244","9571963713","2609850808","394024464","25401600"],["804913564257","2986817524020","4840276154454","4473611753292","2578840102593","949276431864","217859607568","28494372096","1625702400"],["131200910973891","552049255120077","1030897232621622","1121261084297370","782713488739311","363618106703865","112402507014568","22291210864656","2573033630976","131681894400"],["26371383105752091","124081991376524577","262415269268953722","328463201205933570","269451519666338511","151358588321407965","58954714580314668","15720784085252656","2746300846291776","283771423872000","13168189440000"],["6408246094697758113","33342861260291475222","78780831388915228263","111568805474585257872","105220766624838220143","69383770841729095326","32640384829906828089","10953670996934470236","2569565979964472944","401258858402200896","37536212322432000","1593350922240000"],["1851983121367652094657","10558874341860713507430","27569032292878473399975","43587824502158932394880","46474709542918522754895","35203700167236412249806","19424334217052063044665","7865826333620645143020","2319933191768296394800","485981311193120162880","68629240971099777024","5865692990957568000","229442532802560000"],["627822278143634060088723","3892443549401915083015803","11130351711060263065347195","19435438963728340086460095","23121268875914238788644125","19788280269446374098261489","12534274627843603042358649","5949227609779197358032165","2115782002391341507007580","556816373903309825937520","105396154280840131937856","13586811648050477869056","1069083134091896832000","38775788043632640000"],["246734155310448185614868139","1652783481431104903402600287","5137147158129458740952545023","9819676448113049214786867555","12896004705125050500883319745","12308562845571615823191013677","8804472861554025318906200901","4794764277700570758008936049","1997550938456519894428283280","633522107412703696966931040","150556697917418897735331328","25997263216728503662358784","3083164754716009117310976","224779178983159406592000","7600054456551997440000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","52","1456","71344","5422144","591013696","87470027008","16881715212544","4119138511860736","1239860692070081536","451309291913509679104","195416923398549691052032","99271797086463243054432256","58471088483926850159060598784"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","0"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192311,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-aa07d28dcbb1825e"]},{"id":192312,"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-65ac5b4158e0c0e8","e-aa07d28dcbb1825e"]},{"id":192313,"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-65ac5b4158e0c0e8","e-aa07d28dcbb1825e"]},{"id":192314,"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-65ac5b4158e0c0e8","e-aa07d28dcbb1825e"]},{"id":192315,"position":4,"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-65ac5b4158e0c0e8"]}],"region_exact":{},"incoming_edges":[{"id":"e-65ac5b4158e0c0e8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-43f67bf0103635e1","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":"8f9e64e5f29a9e9027d40bbc9d2b1bcfaccecec664bb71f3de35a840b55b26d8","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-6a1d0871f44aed3c","role":"sequence","name":"basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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]"}]},{"id":"e-aa07d28dcbb1825e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-43f67bf0103635e1","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":"7679915442d0814cad5f9c19b8a26df2b6597750d1ae9fd9f87e7016a718b052","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6a1d0871f44aed3c","role":"sequence","name":"basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:               1;\n  1:               3,               1;\n  2:              27,              21,                4;\n  3:             513,             642,              265,               36;\n  4:           16929,           29394,            19017,             5428,             576;\n  5:          863379,         1922319,          1704717,           752253,          165076,           14400;\n  6:        63026667,       171410931,        193647825,        116284281,        39131656,         6993936,          518400;\n  7:      6239640033,     20057988852,      27570270294,      21000887244,      9571963713,      2609850808,       394024464,       25401600;\n  8:    804913564257,   2986817524020,    4840276154454,    4473611753292,   2578840102593,    949276431864,    217859607568,    28494372096,    1625702400;\n  9: 131200910973891, 552049255120077, 1030897232621622, 1121261084297370, 782713488739311, 363618106703865, 112402507014568, 22291210864656, 2573033630976, 131681894400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n^2+1+n^2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n^2 + 1 + n^2*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n^2 + 1 + n^2*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-e734a631f8279507","steps":["basis A007318","basis A007318"],"edges":["e-0b3719baa0dde23e","e-65ac5b4158e0c0e8"],"text":"A008955 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A008955 \u2192 basis A007318 \u2192 basis A007318","chain":null}
