{"id":"n-ebdb57c5e69f5463","revision":1,"first_index":1,"fingerprint":"f6-b1d0161dfb5220fa","name":"basis-map coefficients of Riordan array (1, 3+x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-27","3"],["720","-161","9"],["-19197","6453","-723","27"],["511839","-229653","38638","-2889","81"],["-13646880","7658874","-1719225","192951","-10827","243"],["363859209","-245151522","68818302","-10302713","867564","-38961","729"],["-9701376723","7628091975","-2570416821","481172373","-54041973","3641625","-136323","2187"],["258662438640","-232492390587","91421013852","-20541361815","2884539601","-259231581","14560074","-467289","6561"],["-6896573452773","6974927657271","-3135082702401","821978842383","-138539104542","15566054892","-1165948209","56141019","-1576827","19683"],["183879521277111","-206661673150713","104516734421385","-31322383581357","6159990131559","-830683171880","77788494072","-4994886465","210471048","-5255361","59049"],["-4902677913407040","6061832816532372","-3406746176850654","1148721667676946","-258218053512903","40629782436867","-4566279266811","366556199391","-20597045229","771553917","-17340723","177147"],["130717387970496081","-176333685206961636","109020332941621596","-40849305371175906","10331290153113831","-1858022147655639","243648107895970","-23473112711733","1648899693459","-82365322149","2777074470","-56746089","531441"],["-3485245373982730827","5093702491237254027","-3435827496636479742","1416247002660671082","-398020066098308082","80536582350599040","-12070839060319941","1356853081843371","-114387700252140","7143021747918","-321149356713","9844432767","-184410027","1594323"],["92925168605799267600","-146267982830520900405","106890933201215159115","-48069487480465353690","14861451875887131075","-3341494432279083744","563471546561716158","-72392644268429273","7120713942385029","-533634500357406","29992738882626","-1225959965577","34450839972","-595745361","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-24","568","-13440","318016","-7524864","178052608","-4213063680","99689107456","-2358834069504","55814504808448","-1320677442846720","31249742589853696","-739428402613714944","17496283722010329088"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099097","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["-27","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-27","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160220,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(77/9, 9]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"77/9"},"upper":{"closed":true,"value":"9"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (77/9, 9]","region_exact":null,"edges":["e-0c114df991087062"]},{"id":160221,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-0c114df991087062"]},{"id":160222,"position":2,"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-0c114df991087062"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c114df991087062","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-9\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ebdb57c5e69f5463","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":"7fc57977a67caec6a1eb6914a7a757db94d66a03c6a82d756dd3da12bc6acc6b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"77/9"},"upper":{"closed":true,"value":"9"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4/9"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]"}]}],"used_by":[{"id":"e-2d0268d4f6f3cbee","operation":"basis_map","operation_detail":"A038291","conclusion_id":"n-5d39164c4108e043","conclusion_name":"Riordan array (1, 3+x)."}],"used_by_total":1,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:              1;\n   2:            -27,             3;\n   3:            720,          -161,              9;\n   4:         -19197,          6453,           -723,           27;\n   5:         511839,       -229653,          38638,        -2889,            81;\n   6:      -13646880,       7658874,       -1719225,       192951,        -10827,         243;\n   7:      363859209,    -245151522,       68818302,    -10302713,        867564,      -38961,         729;\n   8:    -9701376723,    7628091975,    -2570416821,    481172373,     -54041973,     3641625,     -136323,     2187;\n   9:   258662438640, -232492390587,    91421013852, -20541361815,    2884539601,  -259231581,    14560074,  -467289,     6561;\n  10: -6896573452773, 6974927657271, -3135082702401, 821978842383, -138539104542, 15566054892, -1165948209, 56141019, -1576827, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(27-3*x)*P(n-1,x) - (9-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -27 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -27 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(27 - 3*x)*P[n - 1, x] - (9 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -27 + 3*x; for(n = 3, N, my(i = n); v[i] = -(27 - 3*x)*v[i - 1] - (9 - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["x \u21a6 x\u22129"],"edges":["e-0c114df991087062"],"text":"A099097 \u2192 x \u21a6 x\u22129"},"headline":"A099097 \u2192 x \u21a6 x\u22129","chain":null}
