{"id":"n-b323b1948005800d","revision":0,"first_index":0,"fingerprint":"f6-e10cc3609cf27a7c","name":"shift_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":[["31","12","1"],["229","129","21","1"],["1961","1432","342","32","1"],["19081","16965","5290","730","45","1"],["207775","216276","82635","14960","1365","60","1"],["2501801","2968357","1335411","297535","35735","2331","77","1"],["32989969","43761264","22556716","5941376","880950","75824","3724","96","1"],["472630861","690761097","399936132","121142532","21296646","2268126","147252","5652","117","1"],["7307593151","11633919580","7453166805","2544545760","515822790","65274552","5252730","266880","8235","140","1"],["121247816845","208356640041","145971392545","55318281645","12671551530","1852830210","177450042","11189970","457545","11605","165","1"],["2148321709561","3955253482632","3001796550786","1247704949920","318013463295","52645686192","5835060924","438479712","22275495","749320","15906","192","1"],["40476722545169","79346477498509","64732502797326","29227949402366","8189216110075","1511228925063","189921445428","16536777972","1002110967","41916875","1180894","21294","221","1"],["807927483311551","1677524800611492","1461680381652127","711276304687312","216946848449331","44087010059092","6185101061139","611357782464","42968914989","2145675532","75216141","1801072","27937","252","1"],["17028146983530961","37281784258845645","34506641729368095","17977546478570315","5921488869317385","1312145696234373","202994041064815","22424154725115","1790829566955","103799991295","4346755413","129583545","2670395","36015","285","1"],["377844723929464865","868958899878025696","850360541740654680","471676166213754880","166654007823359420","39943095521565600","6746959846995368","822772855178560","73501582521510","4844496868640","235628072680","8395890048","215504380","3862880","45720","320","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["44","380","3768","42112","523072","7141248","106209920","1708188416","29525850624","545607622144","10730032423936","223691600732160","4926284479250432","114255071320260608","2783085758131765248"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["31","12","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189973,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-9db1fe647f80bb4e"]},{"id":189974,"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-9db1fe647f80bb4e"]},{"id":189975,"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-9db1fe647f80bb4e"]}],"region_exact":{},"incoming_edges":[{"id":"e-9db1fe647f80bb4e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b323b1948005800d","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":"873adf796453ecb09519ba967bf6d02eb5b722b7837d1e1adc0febf231a75789","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"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:           31,           12,            1;\n  1:          229,          129,           21,           1;\n  2:         1961,         1432,          342,          32,           1;\n  3:        19081,        16965,         5290,         730,          45,          1;\n  4:       207775,       216276,        82635,       14960,        1365,         60,         1;\n  5:      2501801,      2968357,      1335411,      297535,       35735,       2331,        77,        1;\n  6:     32989969,     43761264,     22556716,     5941376,      880950,      75824,      3724,       96,      1;\n  7:    472630861,    690761097,    399936132,   121142532,    21296646,    2268126,    147252,     5652,    117,     1;\n  8:   7307593151,  11633919580,   7453166805,  2544545760,   515822790,   65274552,   5252730,   266880,   8235,   140,   1;\n  9: 121247816845, 208356640041, 145971392545, 55318281645, 12671551530, 1852830210, 177450042, 11189970, 457545, 11605, 165, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (6+2*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 31 + 12*x + x^2.","mathematica":"P[0, x_] := 31 + 12*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + (6 + 2*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 31 + 12*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + (6 + 2*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), 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+1"],"edges":["e-9db1fe647f80bb4e"],"text":"A143497 \u2192 x \u21a6 x+1"},"headline":"A143497 \u2192 x \u21a6 x+1","chain":null}
