{"id":"n-a97ae88764647980","revision":0,"first_index":1,"fingerprint":"f6-4ec588d09c94b2a1","name":"mobius_x_over_x_plus_one of Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","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"],["6","7"],["60","140","81"],["840","2940","3402","1303"],["15120","70560","122472","93816","26785"],["332640","1940400","4490640","5159880","2946350","669351"],["8648640","60540480","175134960","268313760","229815300","104418756","19674097"],["259459200","2118916800","7355668320","14086472400","16087071000","10963969380","4131560370","664400311"],["8821612800","82335052800","333456963840","766304098560","1093920828000","994066557120","561892210320","180716884592","25330978113"],["335221286400","3519823507200","16291754519040","43679333617920","74824184635200","84992690633760","64055711976480","30902587265232","8663194514646","1075631907655"],["14079294028800","164258430336000","855317112249600","2620760017075200","5237692924464000","7139386013235840","6725849757530400","4326362217132480","1819270848075660","451765401215100","50322142646161"],["647647525324800","8311476575001600","48087828755366400","165763071080006400","378610374254112000","602088220449555840","680655995462076480","547284820467258720","306850349708761320","114296646507420300","25463004178957466","2570964805355607"],["32382376266240000","453353267727360000","2885269725321984000","11050871405333760000","28395778069058400000","51607561752819072000","68065599546207648000","65674178456071046400","46027552456314198000","22859329301484060000","7638901253687239800","1542578883213364200","142389639792952801"],["1748648318376960000","26521166162050560000","184132667925093888000","775771172654429952000","2214870689386555200000","4528563543809873568000","6826007268776824128000","7683878879360312428800","6462268364866513399200","4011812292410452530000","1787502893362814113200","541445188007890834200","99957527134652866302","8495817365820681703"],["101421602465863680000","1656552840275773440000","12459643862931353088000","57266017472308829184000","179847499978188282240000","408577066397068593024000","692839737780847648992000","891329950005796241740800","874560318711934813358400","651518316287457490872000","362863087352651264979600","146551164220802452456800","40582756016669063718612","6898603701046393542836","543205619067484203825"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","281","8485","328753","15539261","866545993","55667517781","4046845186145","328340133863533","29405064157989241","2880630399769196933","306375746133297325201","35152976119541661109405","4327595341849511548520873"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["54","269","471","352","96"],["-24","-160","-328","-272","-80"],["0","16","48","48","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["24","200","552","696","416","96"],["0","-32","-128","-192","-128","-32"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","16","80","160","160","80","16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191094,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-88f5365ee0919f31"]},{"id":191095,"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-88f5365ee0919f31"]},{"id":191096,"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-88f5365ee0919f31"]}],"region_exact":{},"incoming_edges":[{"id":"e-88f5365ee0919f31","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-a97ae88764647980","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"5b5ffa3953652221a231e22c378403b06f4561924187fa8b2e2d4851f56101fd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-66d7d2fd3a2f1f0d","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","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":[],"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:            6,             7;\n   3:           60,           140,             81;\n   4:          840,          2940,           3402,           1303;\n   5:        15120,         70560,         122472,          93816,          26785;\n   6:       332640,       1940400,        4490640,        5159880,        2946350,         669351;\n   7:      8648640,      60540480,      175134960,      268313760,      229815300,      104418756,       19674097;\n   8:    259459200,    2118916800,     7355668320,    14086472400,    16087071000,    10963969380,     4131560370,      664400311;\n   9:   8821612800,   82335052800,   333456963840,   766304098560,  1093920828000,   994066557120,   561892210320,   180716884592,   25330978113;\n  10: 335221286400, 3519823507200, 16291754519040, 43679333617920, 74824184635200, 84992690633760, 64055711976480, 30902587265232, 8663194514646, 1075631907655;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(24*n-54-(16*n^2-160*n+269)*x-(48*n^2-328*n+471)*x^2-(48*n^2-272*n+352)*x^3-(16*n^2-80*n+96)*x^4)*P(n-1,x) + (24-(32*n-200)*x-(128*n-552)*x^2-(192*n-696)*x^3-(128*n-416)*x^4-(32*n-96)*x^5)*d/dx P(n-1,x) + (16*x+80*x^2+160*x^3+160*x^4+80*x^5+16*x^6)*d^2/dx^2 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[(-(24*n - 54 - (16*n^2 - 160*n + 269)*x - (48*n^2 - 328*n + 471)*x^2 - (48*n^2 - 272*n + 352)*x^3 - (16*n^2 - 80*n + 96)*x^4)*P[n - 1, x] + (24 - (32*n - 200)*x - (128*n - 552)*x^2 - (192*n - 696)*x^3 - (128*n - 416)*x^4 - (32*n - 96)*x^5)*D[P[n - 1, x], x] + (16*x + 80*x^2 + 160*x^3 + 160*x^4 + 80*x^5 + 16*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];\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] = (-(24*n - 54 - (16*n^2 - 160*n + 269)*x - (48*n^2 - 328*n + 471)*x^2 - (48*n^2 - 272*n + 352)*x^3 - (16*n^2 - 80*n + 96)*x^4)*v[i - 1] + (24 - (32*n - 200)*x - (128*n - 552)*x^2 - (192*n - 696)*x^3 - (128*n - 416)*x^4 - (32*n - 96)*x^5)*deriv(v[i - 1], x) + (16*x + 80*x^2 + 160*x^3 + 160*x^4 + 80*x^5 + 16*x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); 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-66d7d2fd3a2f1f0d","steps":["x \u21a6 x/(x+1)"],"edges":["e-88f5365ee0919f31"],"text":"A292219 \u2192 x \u21a6 x/(x+1)"},"headline":"A292219 \u2192 x \u21a6 x/(x+1)","chain":null}