{"id":"n-4a34fa9bc0fd49a5","revision":0,"first_index":0,"fingerprint":"f6-7299fca8fb81f97e","name":"shift_x_plus_one of veronese_section_2_1 of A133080 * A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["2"],["6","1"],["8","4"],["24","16","1"],["32","32","6"],["96","112","30","1"],["128","192","80","8"],["384","640","320","48","1"],["512","1024","672","160","10"],["1536","3328","2464","720","70","1"],["2048","5120","4608","1792","280","12"],["6144","16384","16128","7168","1400","96","1"],["8192","24576","28160","15360","4032","448","14"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","2","7","12","41","70","239","408","1393","2378","8119","13860","47321","80782"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["2"],["6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171485,"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-76eda9cae6b39812"]},{"id":171486,"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-76eda9cae6b39812"]},{"id":171487,"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-76eda9cae6b39812"]}],"region_exact":{},"incoming_edges":[{"id":"e-76eda9cae6b39812","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4a34fa9bc0fd49a5","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":"1522dd5693c6653b5758a9de2cfbe0748a5e75c0c32ff7c12a8c744ed46d7d17","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-78dea90981c95c84","role":"sequence","name":"veronese_section_2_1 of A133080 * A007318.","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:   0;\n  1:   1;\n  2:   2;\n  3:   6,   1;\n  4:   8,   4;\n  5:  24,  16,   1;\n  6:  32,  32,   6;\n  7:  96, 112,  30,  1;\n  8: 128, 192,  80,  8;\n  9: 384, 640, 320, 48, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) + (4+2*x)*P(n-2,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2, P(3,x) = 6 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 2;\nP[3, x_] := 6 + x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] + (4 + 2*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 2; v[4] = 6 + x; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] + (4 + 2*x)*v[i - 2]); 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-d75669a8cd7f9393","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-7fa3f740a64ca136","e-76eda9cae6b39812"],"text":"A133087 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A133087 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}