{"id":"n-562ff9e6276371c2","revision":0,"first_index":0,"fingerprint":"f6-e38b424e1840fcda","name":"mobius_x_over_x_plus_one of Triangle of unsigned 2-Lah numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["20","50","31"],["120","450","558","229"],["840","4200","7812","6412","1961"],["6720","42000","104160","128240","78440","19081"],["60480","453600","1406160","2308320","2117880","1030374","207775"],["604800","5292000","19686240","40395600","49417200","36063090","14544250","2501801"],["6652800","66528000","288731520","710962560","1087178400","1057850640","639947000","220158488","32989969"],["79833600","898128000","4454714880","12797326080","23483053440","28561967280","23038092000","11888558352","3562916652","472630861"],["1037836800","12972960000","72389116800","237664627200","508799491200","742611149280","748737990000","515170861920","231589582380","61442011930","7307593151"],["14529715200","199783584000","1238658220800","4575044073600","11193588806400","19060352831520","23061130092000","19834078183920","11888265228840","4731034918610","1125369345254","121247816845"],["217945728000","3269185920000","22295847974400","91500881472000","251855748144000","490123358524800","691833902760000","714026814621120","534971935297800","283862095116600","101283241072860","21824607032100","2148321709561"],["3487131648000","56665889280000","421594216243200","1903218334617600","5820666179328000","12743207321644800","20557350253440000","24752929573532160","22254832508388480","14760828946063200","7022304714384960","2269759131338400","446850915588688","40476722545169"],["59281238016000","1037421665280000","8361618622156800","41178723967180800","138531855068006400","336987038061273600","611581170039840000","841599605500093440","882775022832743040","702615457832608320","417827130505905120","180067557752846400","53175258955053872","9633459965750222","807927483311551"],["1067062284288000","20007417830400000","173664386767872000","926521289261568000","3400327351669248000","9098650027654387200","18347435101195200000","28403986685628153600","34049893737834374400","31617695602467374400","22562665047318876480","12154560148317132000","4785773305954848480","1300517095376279970","218140420494118770","17028146983530961"],["20274183401472000","405483668029440000","3770998112673792000","21666343995039744000","86141626242287616000","251453600764266700800","557762027076334080000","959423550270106521600","1293895962037706227200","1373111351878583116800","1143175029064156408320","738997257017681625600","363718771252568484480","131785732331463036960","33157343915106053040","5176556682993412144","377844723929464865"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["101","1357","21225","378641","7584589","168504981","4110999377","109237221145","3139723220661","97043082816989","3209213885373241","113054171838042657","4226238529490065565","167077932827037652261","6964039753216327195809"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["20","50","31"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","-5","-11","-6","0"],["-4","-13","-13","-3","1"],["0","1","3","3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","18","30","22","6","0"],["0","-2","-8","-12","-8","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","5","10","10","5","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189979,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-c15e0629be22c56e"]},{"id":189980,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-c15e0629be22c56e"]},{"id":189981,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-c15e0629be22c56e"]}],"region_exact":{},"incoming_edges":[{"id":"e-c15e0629be22c56e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-562ff9e6276371c2","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":"29dfba10b2da92e27836be0962511b63d64a3e426a4a3401db3a4c46a5fe90e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f3352d3240f9f08b","role":"sequence","name":"Triangle of unsigned 2-Lah numbers.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          20,           50,            31;\n  1:         120,          450,           558,           229;\n  2:         840,         4200,          7812,          6412,           1961;\n  3:        6720,        42000,        104160,        128240,          78440,          19081;\n  4:       60480,       453600,       1406160,       2308320,        2117880,        1030374,         207775;\n  5:      604800,      5292000,      19686240,      40395600,       49417200,       36063090,       14544250,        2501801;\n  6:     6652800,     66528000,     288731520,     710962560,     1087178400,     1057850640,      639947000,      220158488,       32989969;\n  7:    79833600,    898128000,    4454714880,   12797326080,    23483053440,    28561967280,    23038092000,    11888558352,     3562916652,     472630861;\n  8:  1037836800,  12972960000,   72389116800,  237664627200,   508799491200,   742611149280,   748737990000,   515170861920,   231589582380,   61442011930,    7307593151;\n  9: 14529715200, 199783584000, 1238658220800, 4575044073600, 11193588806400, 19060352831520, 23061130092000, 19834078183920, 11888265228840, 4731034918610, 1125369345254, 121247816845;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(4*n-(n^2-13*n-5)*x-(3*n^2-13*n-11)*x^2-(3*n^2-3*n-6)*x^3-(n^2+n)*x^4)*P(n-1,x) + (4-(2*n-18)*x-(8*n-30)*x^2-(12*n-22)*x^3-(8*n-6)*x^4-2*n*x^5)*d/dx P(n-1,x) + (x+5*x^2+10*x^3+10*x^4+5*x^5+x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 20 + 50*x + 31*x^2.","mathematica":"P[0, x_] := 20 + 50*x + 31*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - (n^2 - 13*n - 5)*x - (3*n^2 - 13*n - 11)*x^2 - (3*n^2 - 3*n - 6)*x^3 - (n^2 + n)*x^4)*P[n - 1, x] + (4 - (2*n - 18)*x - (8*n - 30)*x^2 - (12*n - 22)*x^3 - (8*n - 6)*x^4 - 2*n*x^5)*D[P[n - 1, x], x] + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 20 + 50*x + 31*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n - (n^2 - 13*n - 5)*x - (3*n^2 - 13*n - 11)*x^2 - (3*n^2 - 3*n - 6)*x^3 - (n^2 + n)*x^4)*v[i - 1] + (4 - (2*n - 18)*x - (8*n - 30)*x^2 - (12*n - 22)*x^3 - (8*n - 6)*x^4 - 2*n*x^5)*deriv(v[i - 1], x) + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f3352d3240f9f08b","steps":["x \u21a6 x/(x+1)"],"edges":["e-c15e0629be22c56e"],"text":"A143497 \u2192 x \u21a6 x/(x+1)"},"headline":"A143497 \u2192 x \u21a6 x/(x+1)","chain":null}