{"id":"n-fd1f27ef1d8123fd","revision":1,"first_index":0,"fingerprint":"f6-cacf9f85ae71fd07","name":"basis-map coefficients of Riordan array (1, 2+2x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-14","2"],["182","-54","4"],["-2352","1064","-160","8"],["30380","-18480","4204","-424","16"],["-392392","299880","-91448","13912","-1056","32"],["5068168","-4663624","1784216","-363320","41536","-2528","64"],["-65460864","70443968","-32426240","8277248","-1265536","115904","-5888","128"],["845497744","-1041710208","560549024","-172079040","32963856","-4035264","308288","-13440","256"],["-10920516320","15157801120","-9336251456","3349470656","-771380064","118267680","-12072320","791168","-30208","512"],["141050260064","-217775309920","151092015872","-62034887488","16692610144","-3076086240","393161280","-34416256","1974784","-67072","1024"],["-1821816412416","3096904610688","-2389815719424","1105107364480","-340268054784","73251920000","-11250882560","1232929152","-94474240","4820992","-147456","2048"],["23530726132928","-43669342515456","37100390451264","-19080462084992","6616201178944","-1629612561920","292359765440","-38494623104","3692018176","-251553792","11557888","-321536","4096"],["-303924736087168","611431950107776","-567092922055296","321076115550848","-123813773179264","34340915235200","-7048245644160","1083881481088","-124889003008","10649347072","-653211648","27293696","-696320","8192"],["3925516139359360","-8509464526200960","8555420657641088","-5287924211730048","2244757314936192","-692353381426560","160005007648640","-28147187769216","3787531503616","-387963075584","29778739200","-1660917760","63635456","-1499136","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-12","132","-1440","15696","-171072","1864512","-20321280","221481216","-2413919232","26309256192","-286744043520","3125217447936","-34061680852992","371237560860672"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099040","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-14","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-14","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-14","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160116,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(5, 7]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"5"},"upper":{"closed":true,"value":"7"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (5, 7]","region_exact":null,"edges":["e-5e00209341210844"]},{"id":160117,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-5e00209341210844"]},{"id":160118,"position":2,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-5e00209341210844"]}],"region_exact":{},"incoming_edges":[{"id":"e-5e00209341210844","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-7\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fd1f27ef1d8123fd","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":"3c805243d2abd1d08f2cfb5fa614aacaa53f18120fba8c0af7eaca24bc2f5121","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"5"},"upper":{"closed":true,"value":"7"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]"}]}],"used_by":[{"id":"e-3f03a2606d2e6662","operation":"basis_map","operation_detail":"A027466","conclusion_id":"n-f5a001953ee28e8f","conclusion_name":"Riordan array (1, 2+2x)."}],"used_by_total":1,"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:          -14,           2;\n  2:          182,         -54,           4;\n  3:        -2352,        1064,        -160,          8;\n  4:        30380,      -18480,        4204,       -424,         16;\n  5:      -392392,      299880,      -91448,      13912,      -1056,        32;\n  6:      5068168,    -4663624,     1784216,    -363320,      41536,     -2528,        64;\n  7:    -65460864,    70443968,   -32426240,    8277248,   -1265536,    115904,     -5888,    128;\n  8:    845497744, -1041710208,   560549024, -172079040,   32963856,  -4035264,    308288, -13440,    256;\n  9: -10920516320, 15157801120, -9336251456, 3349470656, -771380064, 118267680, -12072320, 791168, -30208, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(14-2*x)*P(n-1,x) - (14-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -14 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -14 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(14 - 2*x)*P[n - 1, x] - (14 - 2*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -14 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(14 - 2*x)*v[i - 1] - (14 - 2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f5a001953ee28e8f","steps":["x \u21a6 x\u22127"],"edges":["e-5e00209341210844"],"text":"A099040 \u2192 x \u21a6 x\u22127"},"headline":"A099040 \u2192 x \u21a6 x\u22127","chain":null}