{"id":"n-6379632195c99c4b","revision":1,"first_index":0,"fingerprint":"f6-7eb4d7879a2f4529","name":"shift_x_plus_one of Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","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":[["1"],["3","1"],["11","7","1"],["43","41","12","1"],["171","219","99","18","1"],["683","1101","678","195","25","1"],["2731","5311","4149","1644","340","33","1"],["10923","24881","23568","11925","3430","546","42","1"],["43691","114083","127031","78278","28915","6454","826","52","1"],["174763","514581","658602","478599","216143","62271","11256","1194","63","1"],["699051","2291335","3314025","2774880","1482110","523887","122787","18516","1665","75","1"],["2796203","10097401","16285300","15439065","9522740","3991400","1151458","226083","29073","2255","88","1"],["11184811","44117867","78500811","83118970","58149845","28203612","9671284","2344232","394020","43945","2981","102","1"],["44739243","191384541","372412782","435601611","340854645","187822125","74669472","21562164","4486482","656370","64350","3861","117","1"],["178956971","825190479","1743173469","2232236628","1931900376","1192287117","539342181","180884088","44932446","8158722","1052766","91728","4914","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","97","508","2683","14209","75316","399331","2117473","11228332","59541067","315732481","1674257764","8878212019"],"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":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["-11","-5"],["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":164695,"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-822f6e0d1df19a88"]}],"region_exact":{},"incoming_edges":[{"id":"e-822f6e0d1df19a88","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6379632195c99c4b","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":"1ec9b561e94b9f9974cfc75a6356870cb14ac384e295c10289df16b1aae47199","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fac4d4a000d744e8","role":"sequence","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","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:      1;\n  1:      3,      1;\n  2:     11,      7,      1;\n  3:     43,     41,     12,      1;\n  4:    171,    219,     99,     18,      1;\n  5:    683,   1101,    678,    195,     25,     1;\n  6:   2731,   5311,   4149,   1644,    340,    33,     1;\n  7:  10923,  24881,  23568,  11925,   3430,   546,    42,    1;\n  8:  43691, 114083, 127031,  78278,  28915,  6454,   826,   52,  1;\n  9: 174763, 514581, 658602, 478599, 216143, 62271, 11256, 1194, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (14*n-11+(6*n-5)*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n - 11 + (6*n - 5)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\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] = ((14*n - 11 + (6*n - 5)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 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-fac4d4a000d744e8","steps":["x \u21a6 x+1"],"edges":["e-822f6e0d1df19a88"],"text":"A060920 \u2192 x \u21a6 x+1"},"headline":"A060920 \u2192 x \u21a6 x+1","chain":null}