{"id":"n-8910fc24d8983662","revision":1,"first_index":0,"fingerprint":"f6-6a7f301aedff1f06","name":"basis-map coefficients of Riordan array (1, 3+3x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-45","3"],["1980","-267","9"],["-87075","17685","-1197","27"],["3829275","-1039095","105714","-4779","81"],["-168399000","57190050","-7767495","527391","-17901","243"],["7405637625","-3020502150","513233010","-46502883","2369736","-64395","729"],["-325675738125","155060760375","-31635884475","3585293685","-243759051","9942345","-225261","2187"],["14322154522500","-7796621921625","1856640090600","-252613540485","21478891581","-1168675695","39737790","-772011","6561"],["-629841545296875","385859688609375","-105049872759375","16681283724375","-1702665704250","115848398340","-5254263855","153179667","-2604717","19683"],["27698372584846875","-18859396173271875","5777884670158125","-1048869856281375","124939417121775","-10204147957032","578692840860","-22501922715","574139988","-8680203","59049"],["-1218083896779750000","912514734928462500","-310698175336931250","63466990157261250","-8642219531459625","823683734393715","-56069634641301","2726009168175","-92765316339","2104329213","-28638765","177147"],["53567348588770640625","-43785646816568287500","16402379096270362500","-3723576285544413750","570531966048142875","-62158421732664285","4937531140329894","-288130651447023","12259124672175","-370877734503","7573034250","-93710763","531441"],["-2355716911139590078125","2086338737749764796875","-852745037686923881250","212979961055222156250","-36265366479412203750","4445732449472225400","-403669981750798395","27487593419613939","-1403700097849164","53093881305558","-1445818012695","26842199175","-304515693","1594323"],["103596730314786874687500","-98821337779646301234375","43763078909378997871875","-11925565290407458968750","2234036707410415966875","-304345334788372432800","31093682360689765890","-2420173176398916723","144213232097465253","-6546858080228442","222890034743190","-5518416913875","93924756576","-983697291","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-42","1722","-70560","2891196","-118466712","4854171672","-198899608320","8149908339216","-333942366697632","13683283251053472","-560672317142945280","22973539423459455936","-941340418465293447552","38571408919757027647872"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099093","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-45","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-45","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-45","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160160,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(41/3, 15]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"41/3"},"upper":{"closed":true,"value":"15"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (41/3, 15]","region_exact":null,"edges":["e-b320272c2500474a"]},{"id":160161,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-b320272c2500474a"]},{"id":160162,"position":2,"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-b320272c2500474a"]}],"region_exact":{},"incoming_edges":[{"id":"e-b320272c2500474a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-15\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8910fc24d8983662","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":"ee76d124510616a9b77df9e17a4a91569a7b0954a26df9ed9de3972dec30b006","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"41/3"},"upper":{"closed":true,"value":"15"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4/3"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]"}]}],"used_by":[{"id":"e-35fdacf74fcfdf8c","operation":"basis_map","operation_detail":"A027467","conclusion_id":"n-016d7724d51f6416","conclusion_name":"Riordan array (1, 3+3x)."}],"used_by_total":1,"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:              -45,               3;\n  2:             1980,            -267,                9;\n  3:           -87075,           17685,            -1197,             27;\n  4:          3829275,        -1039095,           105714,          -4779,             81;\n  5:       -168399000,        57190050,         -7767495,         527391,         -17901,          243;\n  6:       7405637625,     -3020502150,        513233010,      -46502883,        2369736,       -64395,         729;\n  7:    -325675738125,    155060760375,     -31635884475,     3585293685,     -243759051,      9942345,     -225261,      2187;\n  8:   14322154522500,  -7796621921625,    1856640090600,  -252613540485,    21478891581,  -1168675695,    39737790,   -772011,     6561;\n  9: -629841545296875, 385859688609375, -105049872759375, 16681283724375, -1702665704250, 115848398340, -5254263855, 153179667, -2604717, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(45-3*x)*P(n-1,x) - (45-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -45 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -45 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(45 - 3*x)*P[n - 1, x] - (45 - 3*x)*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] = -45 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(45 - 3*x)*v[i - 1] - (45 - 3*x)*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}}],"derivation":{"root":"n-016d7724d51f6416","steps":["x \u21a6 x\u221215"],"edges":["e-b320272c2500474a"],"text":"A099093 \u2192 x \u21a6 x\u221215"},"headline":"A099093 \u2192 x \u21a6 x\u221215","chain":null}
