{"id":"n-7b8be07dc123ab8a","revision":0,"first_index":0,"fingerprint":"f6-05a7fe04d09eab85","name":"theta_plus_one of veronese_section_2_0 of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","6"],["1","18"],["1","36","15"],["1","60","75"],["1","90","225","28"],["1","126","525","196"],["1","168","1050","784","45"],["1","216","1890","2352","405"],["1","270","3150","5880","2025","66"],["1","330","4950","12936","7425","726"],["1","396","7425","25872","22275","4356","91"],["1","468","10725","48048","57915","18876","1183"],["1","546","15015","84084","135135","66066","8281","120"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","19","52","136","344","848","2048","4864","11392","26368","60416","137216","309248"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1","6"],["1","18"],["1","36","15"],["1","60","75"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","3","-3","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["6","-12","6"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["-15","18","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["20","-12"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-15","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172821,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-675e8768a154ef5a"]},{"id":172822,"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-675e8768a154ef5a"]},{"id":172823,"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-675e8768a154ef5a"]}],"region_exact":{},"incoming_edges":[{"id":"e-675e8768a154ef5a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-7b8be07dc123ab8a","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"098aa378bd91f5e4c80f1ede36a5876cc49964e90600998851b64c4e616f2bee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eba9e40af29eafd3","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","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: 1;\n  2: 1,   6;\n  3: 1,  18;\n  4: 1,  36,   15;\n  5: 1,  60,   75;\n  6: 1,  90,  225,   28;\n  7: 1, 126,  525,  196;\n  8: 1, 168, 1050,  784,  45;\n  9: 1, 216, 1890, 2352, 405;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-3*x+3*x^2-x^3)*P(n-6,x) + (6-12*x+6*x^2)*P(n-5,x) - (15-18*x+3*x^2)*P(n-4,x) + (20-12*x)*P(n-3,x) - (15-3*x)*P(n-2,x) + 6*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 6*x, P(3,x) = 1 + 18*x, P(4,x) = 1 + 36*x + 15*x^2, P(5,x) = 1 + 60*x + 75*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 6*x;\nP[3, x_] := 1 + 18*x;\nP[4, x_] := 1 + 36*x + 15*x^2;\nP[5, x_] := 1 + 60*x + 75*x^2;\nP[n_, x_] := P[n, x] = Expand[-(1 - 3*x + 3*x^2 - x^3)*P[n - 6, x] + (6 - 12*x + 6*x^2)*P[n - 5, x] - (15 - 18*x + 3*x^2)*P[n - 4, x] + (20 - 12*x)*P[n - 3, x] - (15 - 3*x)*P[n - 2, x] + 6*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 6*x; v[4] = 1 + 18*x; v[5] = 1 + 36*x + 15*x^2; v[6] = 1 + 60*x + 75*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(1 - 3*x + 3*x^2 - x^3)*v[i - 6] + (6 - 12*x + 6*x^2)*v[i - 5] - (15 - 18*x + 3*x^2)*v[i - 4] + (20 - 12*x)*v[i - 3] - (15 - 3*x)*v[i - 2] + 6*v[i - 1]); 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-801895951aa20f29","steps":["Veronese 2","\u03b8+1"],"edges":["e-6ce79ceb855f1a6c","e-675e8768a154ef5a"],"text":"A103406 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A103406 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}