{"id":"n-cda2c26e87214b18","revision":0,"first_index":1,"fingerprint":"f6-72244d199a450cc9","name":"reversal of basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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"],["9","4"],["45","60","16"],["189","504","336","64"],["729","3240","3888","1728","256"],["2673","17820","33264","25344","8448","1024"],["9477","88452","235872","269568","149760","39936","4096"],["32805","408240","1469664","2332800","1900800","829440","184320","16384"],["111537","1784592","8328096","17449344","19388160","12220416","4386816","835584","65536"],["373977","7479540","43879968","117013248","169019136","143410176","73543680","22413312","3735552","262144"],["1240029","30311820","218245104","720555264","1307674368","1426553856","975421440","421134336","111476736","16515072","1048576"],["4074381","119515176","1035798192","4143192768","9207095040","12499329024","10896850944","6226771968","2319777792","542638080","72351744","4194304"],["13286025","460582200","4728643920","22517352000","60046272000","98985369600","106599628800","77157826560","37822464000","12386304000","2595225600","314572800","16777216"],["43046721","1741000716","20892008592","116729952768","367483184640","721603344384","937467592704","833304526848","514688090112","220723937280","64465403904","12230590464","1358954496","67108864"],["138706101","6472951380","89758259136","581291582976","2131402470912","4908684478464","7551822274560","8055277092864","6080944472064","3271619248128","1246331142144","328414003200","56925093888","5838471168","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","121","1093","9841","88573","797161","7174453","64570081","581130733","5230176601","47071589413","423644304721","3812798742493","34315188682441"],"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"],["2"]]},"terms":[{"coeff":{"coeffs":[["-3","-4"],["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185559,"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-daa1f1445a2806de"]},{"id":185560,"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-daa1f1445a2806de"]},{"id":185561,"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-daa1f1445a2806de"]}],"region_exact":{},"incoming_edges":[{"id":"e-daa1f1445a2806de","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-cda2c26e87214b18","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":"d3be607ee86e4c84b1439fc98f0b02d07b5cbf36c6dad63f2d4af079a75e8e91","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-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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-47411a88ffe48d9a","kind":"fingerprint_related","node_a_id":"n-194cb2d73cc003c5","node_b_id":"n-cda2c26e87214b18","fingerprint":"f6-72244d199a450cc9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72244d199a450cc9"},"other_id":"n-194cb2d73cc003c5","other":{"id":"n-194cb2d73cc003c5","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","first_index":1,"fingerprint":"f6-72244d199a450cc9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","9"],["16","60","45"]],"card_rows":[["1"],["4","9"],["16","60","45"],["64","336","504","189"]],"row_sums_preview":["1","13","121","1093","9841","88573","797161","7174453"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013610"],"edges":["e-4aa0aeb00b1923c9"],"text":"A091042 \u2192 basis A013610"},"headline":"A091042 \u2192 basis A013610","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:      9,       4;\n   3:     45,      60,       16;\n   4:    189,     504,      336,        64;\n   5:    729,    3240,     3888,      1728,       256;\n   6:   2673,   17820,    33264,     25344,      8448,      1024;\n   7:   9477,   88452,   235872,    269568,    149760,     39936,     4096;\n   8:  32805,  408240,  1469664,   2332800,   1900800,    829440,   184320,    16384;\n   9: 111537, 1784592,  8328096,  17449344,  19388160,  12220416,  4386816,   835584,   65536;\n  10: 373977, 7479540, 43879968, 117013248, 169019136, 143410176, 73543680, 22413312, 3735552, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = (6*n-3+(4*n-4)*x)*P(n-1,x) + (12*x+4*x^2)*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[((6*n - 3 + (4*n - 4)*x)*P[n - 1, x] + (12*x + 4*x^2)*D[P[n - 1, x], x])/(2*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] = ((6*n - 3 + (4*n - 4)*x)*v[i - 1] + (12*x + 4*x^2)*deriv(v[i - 1], x))/(2*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-ba8de4e02acfb6ca","steps":["basis A013610","reverse"],"edges":["e-4aa0aeb00b1923c9","e-daa1f1445a2806de"],"text":"A091042 \u2192 basis A013610 \u2192 reverse"},"headline":"A091042 \u2192 basis A013610 \u2192 reverse","chain":null}