{"id":"n-fc2a1f4caddbe7bf","revision":0,"first_index":1,"fingerprint":"f6-528339e5ff5f0c6c","name":"shift_x_plus_one of (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).","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"],["4","1"],["13","8","1"],["34","39","12","1"],["73","136","78","16","1"],["136","365","340","130","20","1"],["229","816","1095","680","195","24","1"],["358","1603","2856","2555","1190","273","28","1"],["529","2864","6412","7616","5110","1904","364","32","1"],["748","4761","12888","19236","17136","9198","2856","468","36","1"],["1021","7480","23805","42960","48090","34272","15330","4080","585","40","1"],["1354","11231","41140","87285","118140","105798","62832","24090","5610","715","44","1"],["1753","16248","67386","164560","261855","283536","211596","107712","36135","7480","858","48","1"],["2224","22789","105612","292006","534820","680823","614328","392964","175032","52195","9724","1014","52","1"],["2773","31136","159523","492856","1022021","1497496","1588587","1228656","687687","272272","73073","12376","1183","56","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","22","86","304","992","3040","8864","24832","67328","177664","458240","1159168","2883584","7069696"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","1"],["-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172645,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-0d1234184cdf5687"]},{"id":172646,"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-0d1234184cdf5687"]},{"id":172647,"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-0d1234184cdf5687"]}],"region_exact":{},"incoming_edges":[{"id":"e-0d1234184cdf5687","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fc2a1f4caddbe7bf","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":"54417459f9d7a04ca658c8e5081b357102e6eee4b38e76a59c6ab450dc7d4f58","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-77dad805f3db7972","role":"sequence","name":"(I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).","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:   4,    1;\n   3:  13,    8,     1;\n   4:  34,   39,    12,     1;\n   5:  73,  136,    78,    16,     1;\n   6: 136,  365,   340,   130,    20,    1;\n   7: 229,  816,  1095,   680,   195,   24,    1;\n   8: 358, 1603,  2856,  2555,  1190,  273,   28,   1;\n   9: 529, 2864,  6412,  7616,  5110, 1904,  364,  32,  1;\n  10: 748, 4761, 12888, 19236, 17136, 9198, 2856, 468, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-6-x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n - 6 - x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], 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] = -(n - 6 - x)*v[i - 1] + (1 + x)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["D+1","(1/n)D\u00b2+(2/n)D+1/n","x \u21a6 x+1"],"edges":["e-9e777fb0c714fbf5","e-23b6014ad5393438","e-0d1234184cdf5687"],"text":"A132440 \u2192 D+1 \u2192 (1/n)D\u00b2+(2/n)D+1/n \u2192 x \u21a6 x+1"},"headline":"A132440 \u2192 D+1 \u2192 (1/n)D\u00b2+(2/n)D+1/n \u2192 x \u21a6 x+1","chain":null}