{"id":"n-90e0961665129420","revision":0,"first_index":1,"fingerprint":"f6-50f0520d3a512303","name":"reversal of basis_map_A007318 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":1,"rr_by_factors":false,"rows":[["1"],["3","3"],["9","19","10"],["27","87","93","33"],["81","351","568","407","109"],["243","1323","2871","3105","1674","360"],["729","4779","13014","18847","15312","6618","1189"],["2187","16767","54945","99777","108453","70569","25455","3927"],["6561","57591","220644","481959","656551","571225","310002","95953","12970"],["19683","194643","853659","2179521","3570252","3891558","2822703","1313889","356151","42837"],["59049","649539","3209058","9377775","17951922","23523940","21370559","13291003","5416075","1305887","141481"],["177147","2145447","11790117","38808801","85022271","130177359","142145307","110698947","60257826","21835926","4741092","467280"],["531441","7026831","42515280","155655351","384080049","672928587","858443860","803427261","547531881","264988334","86453016","17072484","1543321"],["1594323","22851963","150948927","608447457","1669805118","3294856980","4809316971","5257936029","4305721680","2608517418","1136417802","337153518","61055787","5097243"],["4782969","73870299","528960942","2327731263","7032928356","15433721694","25369530489","31733131447","30352844248","22093676436","12047325875","4772155310","1298153415","217074895","16835050"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","38","240","1516","9576","60488","382080","2413456","15244896","96296288","608267520","3842197696","24269721216","153302722688"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["3"]]},"terms":[{"coeff":{"coeffs":[["-9","-31","-26"],["9","20","13"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-9","-22","-13"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181370,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b97a10781ad79632"]},{"id":181371,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b97a10781ad79632"]},{"id":181372,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b97a10781ad79632"]}],"region_exact":{},"incoming_edges":[{"id":"e-b97a10781ad79632","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-90e0961665129420","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"381b47985ecee03b3603108acd48a1e95f154b6a4862be23a9877cb74579eb14","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a9307f854ca152e2","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0e2be45ab570e1b3","kind":"fingerprint_related","node_a_id":"n-a9307f854ca152e2","node_b_id":"n-90e0961665129420","fingerprint":"f6-50f0520d3a512303","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-50f0520d3a512303"},"other_id":"n-a9307f854ca152e2","other":{"id":"n-a9307f854ca152e2","name":"basis_map_A007318 of Riordan array (1, 3+x).","first_index":1,"fingerprint":"f6-50f0520d3a512303","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","3"],["10","19","9"]],"card_rows":[["1"],["3","3"],["10","19","9"],["33","93","87","27"]],"row_sums_preview":["1","6","38","240","1516","9576","60488","382080"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A007318"],"edges":["e-0e2d8787d6ee9a58"],"text":"A099097 \u2192 basis A007318"},"headline":"A099097 \u2192 basis A007318","chain":null}}],"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:     3,      3;\n   3:     9,     19,     10;\n   4:    27,     87,     93,      33;\n   5:    81,    351,    568,     407,     109;\n   6:   243,   1323,   2871,    3105,    1674,     360;\n   7:   729,   4779,  13014,   18847,   15312,    6618,    1189;\n   8:  2187,  16767,  54945,   99777,  108453,   70569,   25455,    3927;\n   9:  6561,  57591, 220644,  481959,  656551,  571225,  310002,   95953,  12970;\n  10: 19683, 194643, 853659, 2179521, 3570252, 3891558, 2822703, 1313889, 356151, 42837;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3)*P(n,x) = (9*n-9+(20*n-31)*x+(13*n-26)*x^2)*P(n-1,x) - (9*x+22*x^2+13*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 + (20*n - 31)*x + (13*n - 26)*x^2)*P[n - 1, x] - (9*x + 22*x^2 + 13*x^3)*D[P[n - 1, x], x])/(3*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((9*n - 9 + (20*n - 31)*x + (13*n - 26)*x^2)*v[i - 1] - (9*x + 22*x^2 + 13*x^3)*deriv(v[i - 1], x))/(3*n - 3)); 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":["basis A007318","reverse"],"edges":["e-0e2d8787d6ee9a58","e-b97a10781ad79632"],"text":"A099097 \u2192 basis A007318 \u2192 reverse"},"headline":"A099097 \u2192 basis A007318 \u2192 reverse","chain":null}
