{"id":"n-24403ee2e7d633a4","revision":1,"first_index":0,"fingerprint":"f6-1f700cdd56f446cf","name":"shift_x_plus_one of Coefficients of a generalized Jaco-Lucas polynomial (even indices) 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":false,"rows":[["2"],["5","2"],["17","12","2"],["65","66","21","2"],["257","344","164","32","2"],["1025","1710","1095","330","45","2"],["4097","8196","6606","2712","585","60","2"],["16385","38234","37177","19362","5754","952","77","2"],["65537","174768","199048","125696","47700","10976","1456","96","2"],["262145","786438","1026747","762186","352251","104094","19362","2124","117","2"],["1048577","3495260","5145810","4390680","2392995","864572","207570","32160","2985","140","2"],["4194305","15379122","25204685","24302850","15261840","6521284","1920919","385902","50919","4070","165","2"],["16777217","67108872","121168812","130282400","92634390","45709152","15965600","3947856","678249","77528","5412","192","2"],["67108865","290805086","573532271","680340362","540273305","302379194","122215652","35921912","7618754","1138280","114257","7046","221","2"],["268435457","1252698804","2679383574","3475852632","3049211277","1908786012","876503250","298677888","75467574","13957944","1837836","163800","9009","252","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","7","31","154","799","4207","22258","117943","625279","3315466","17580751","93226063","494355682","2621455207","13900985311"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["-14","-6"],["14","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-18","-17","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164947,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-895e752df1147f88"]}],"region_exact":{},"incoming_edges":[{"id":"e-895e752df1147f88","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-24403ee2e7d633a4","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":"f5f2a80a1a44f6aae34a55266f9cd01d847d47783c38a9dbc234d5edd10f1df0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-9b40f045d129e237","role":"sequence","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:      2;\n  1:      5,      2;\n  2:     17,     12,       2;\n  3:     65,     66,      21,      2;\n  4:    257,    344,     164,     32,      2;\n  5:   1025,   1710,    1095,    330,     45,      2;\n  6:   4097,   8196,    6606,   2712,    585,     60,     2;\n  7:  16385,  38234,   37177,  19362,   5754,    952,    77,    2;\n  8:  65537, 174768,  199048, 125696,  47700,  10976,  1456,   96,   2;\n  9: 262145, 786438, 1026747, 762186, 352251, 104094, 19362, 2124, 117, 2;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (14*n-14+(6*n-6)*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n - 14 + (6*n - 6)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((14*n - 14 + (6*n - 6)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 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-9b40f045d129e237","steps":["x \u21a6 x+1"],"edges":["e-895e752df1147f88"],"text":"A122076 \u2192 x \u21a6 x+1"},"headline":"A122076 \u2192 x \u21a6 x+1","chain":null}