{"id":"n-8fd7e451a6f8a0e6","revision":0,"first_index":0,"fingerprint":"f6-0a560bfbcc45e3fa","name":"shift_x_plus_one of Triangle of 4-Eulerian numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5","4"],["30","45","16"],["210","450","305","64"],["1680","4620","4530","1845","256"],["15120","50400","64050","38250","10505","1024"],["151200","589680","915600","716100","292530","57645","4096"],["1663200","7408800","13547520","13053600","7060410","2101050","308705","16384"],["19958400","99792000","209563200","239410080","160912080","64130220","14458530","1625445","65536"],["259459200","1437004800","3402907200","4489732800","3602899440","1800338400","550121250","96503850","8454905","262144"],["3632428800","22054032000","58078944000","86819040000","80926322400","48629140560","18803677200","4526375700","629752530","43585245","1048576"],["54486432000","359610451200","1041728688000","1738775808000","1845198986400","1295554629600","606895359840","186640423200","36092897610","4040026650","223169105","4194304"],["871782912000","6211453248000","19618747948800","36146558448000","43007697532800","34548289776000","19024499894400","7141822047360","1781866566480","280930469820","25579174530","1136817045","16777216"],["14820309504000","113331778560000","387453017952000","780714289555200","1029005327750400","930671847705600","589300686374400","261663219417600","80313197524560","16502376290400","2145567510450","160295949450","5767971305","67108864"],["266765571072000","2178585497088000","8012556744192000","17522509612608000","25336074642278400","25475803280928000","18239670408960000","9360602963712000","3419764169421600","871340164334640","149183550406800","16141044219300","996383524530","29175400845","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","91","1029","12931","179349","2726851","45159669","809915491","15647683989","324144347011","7169251065909","168660381612451","4206087749174229","110850023476581571"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","3"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183620,"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-4590d7badc97de8b"]},{"id":183621,"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-4590d7badc97de8b"]},{"id":183622,"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-4590d7badc97de8b"]},{"id":183623,"position":3,"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-4590d7badc97de8b"]}],"region_exact":{},"incoming_edges":[{"id":"e-4590d7badc97de8b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8fd7e451a6f8a0e6","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":"c811091eaf41438230f30e1d4c0a1615b2d35eb0a2944443cd3a284147f5338e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7e652d8a57b6dcff","role":"sequence","name":"Triangle of 4-Eulerian 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:         1;\n  1:         5,          4;\n  2:        30,         45,         16;\n  3:       210,        450,        305,         64;\n  4:      1680,       4620,       4530,       1845,        256;\n  5:     15120,      50400,      64050,      38250,      10505,       1024;\n  6:    151200,     589680,     915600,     716100,     292530,      57645,      4096;\n  7:   1663200,    7408800,   13547520,   13053600,    7060410,    2101050,    308705,    16384;\n  8:  19958400,   99792000,  209563200,  239410080,  160912080,   64130220,  14458530,  1625445,   65536;\n  9: 259459200, 1437004800, 3402907200, 4489732800, 3602899440, 1800338400, 550121250, 96503850, 8454905, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+4+(n+3)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 4 + (n + 3)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + (n + 3)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 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-7e652d8a57b6dcff","steps":["x \u21a6 x+1"],"edges":["e-4590d7badc97de8b"],"text":"A144698 \u2192 x \u21a6 x+1"},"headline":"A144698 \u2192 x \u21a6 x+1","chain":null}