{"id":"n-ff2c124f57c19690","revision":0,"first_index":0,"fingerprint":"f6-ce6d9281d03644ef","name":"basis_map_A008288 of Another version of Fibonacci-Pascal triangle A037027.","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"],["1","1"],["3","7","2"],["5","21","17","3"],["11","63","91","40","5"],["21","157","345","273","82","8"],["43","391","1155","1383","722","163","13"],["85","917","3441","5681","4469","1721","311","21"],["171","2127","9675","20583","22291","12760","3864","581","34"],["341","4813","25801","67729","95189","74701","33268","8276","1065","55"],["683","10775","66515","208423","363923","370271","224506","81330","17133","1926","89"],["1365","23813","166497","608049","1277685","1618853","1268721","621671","189166","34518","3444","144"],["2731","52191","407707","1702823","4200787","6426919","6266947","3952592","1616847","423203","68059","6103","233"],["5461","113469","979769","4611345","13101013","23630141","27846249","21804385","11431566","4000612","917445","131825","10733","377"],["10923","245159","2318883","12151527","39146067","81661487","113647371","107458447","69748538","31155693","9506812","1938061","251577","18755","610"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","12","46","210","886","3870","16646","72086","311238","1345574","5813926","25127142","108584390","469259910"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008288","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"],["3","7","2"],["5","21","17","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["2","6","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198262,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [\u22123\u22122\u221a2, \u22123+2\u221a2] (rational outer bound [-1564556493/268435456, -46056243/268435456])","region_exact":"[\u22123\u22122\u221a2, \u22123+2\u221a2]","edges":["e-a9970c74218754b4"]},{"id":198263,"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-a9970c74218754b4"]},{"id":198264,"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-a9970c74218754b4"]}],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"incoming_edges":[{"id":"e-a9970c74218754b4","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-ff2c124f57c19690","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"cef378a6c949e14cb0a94aa46aabda6a0990a8d2d2ff6f53950e5c0a0422e3c1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-e7c079f81f452bf3","role":"sequence","name":"Another version of Fibonacci-Pascal triangle A037027.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 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,    1;\n  2:   3,    7,     2;\n  3:   5,   21,    17,     3;\n  4:  11,   63,    91,    40,     5;\n  5:  21,  157,   345,   273,    82,     8;\n  6:  43,  391,  1155,  1383,   722,   163,    13;\n  7:  85,  917,  3441,  5681,  4469,  1721,   311,   21;\n  8: 171, 2127,  9675, 20583, 22291, 12760,  3864,  581,   34;\n  9: 341, 4813, 25801, 67729, 95189, 74701, 33268, 8276, 1065, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-4,x) + (3*x+x^2)*P(n-3,x) + (2+6*x+x^2)*P(n-2,x) + (1+x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 7*x + 2*x^2, P(3,x) = 5 + 21*x + 17*x^2 + 3*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 3 + 7*x + 2*x^2;\nP[3, x_] := 5 + 21*x + 17*x^2 + 3*x^3;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 4, x] + (3*x + x^2)*P[n - 3, x] + (2 + 6*x + x^2)*P[n - 2, x] + (1 + x)*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 + x; v[3] = 3 + 7*x + 2*x^2; v[4] = 5 + 21*x + 17*x^2 + 3*x^3; for(n = 4, N, my(i = n + 1); v[i] = 2*x*v[i - 4] + (3*x + x^2)*v[i - 3] + (2 + 6*x + x^2)*v[i - 2] + (1 + x)*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-e7c079f81f452bf3","steps":["basis A008288"],"edges":["e-a9970c74218754b4"],"text":"A111006 \u2192 basis A008288"},"headline":"A111006 \u2192 basis A008288","chain":null}
