{"id":"n-22333aafcaf54c94","revision":0,"first_index":0,"fingerprint":"f6-92bdcb9f4d02545f","name":"mobius_x_over_x_plus_one of Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","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","2"],["0","1","2"],["0","1","4","4"],["0","0","2","7","6"],["0","0","1","7","15","10"],["0","0","0","3","16","28","16"],["0","0","0","1","11","38","53","26"],["0","0","0","0","4","30","82","97","42"],["0","0","0","0","1","16","79","173","176","68"],["0","0","0","0","0","5","50","191","352","315","110"],["0","0","0","0","0","1","22","145","443","701","559","178"],["0","0","0","0","0","0","6","77","386","986","1368","984","288"],["0","0","0","0","0","0","1","29","244","974","2130","2628","1721","466"],["0","0","0","0","0","0","0","7","112","707","2346","4484","4980","2993","754"],["0","0","0","0","0","0","0","1","37","385","1925","5450","9242","9329","5180","1220"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","3","9","15","33","63","129","255","513","1023","2049","4095","8193","16383","32769"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","2"],["0","1","2"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","1","2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171246,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-68f8f8a28cba92c4"]},{"id":171247,"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-68f8f8a28cba92c4"]},{"id":171248,"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-68f8f8a28cba92c4"]}],"region_exact":{},"incoming_edges":[{"id":"e-68f8f8a28cba92c4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":1,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-22333aafcaf54c94","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"aa6358de82db1705c3c71ac283767f5eabefef0c507f3125fb62bbfa336e6e29","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f239c3b0ba932394","role":"sequence","name":"Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","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, 2;\n  1: 0, 1, 2;\n  2: 0, 1, 4, 4;\n  3: 0, 0, 2, 7,  6;\n  4: 0, 0, 1, 7, 15, 10;\n  5: 0, 0, 0, 3, 16, 28, 16;\n  6: 0, 0, 0, 1, 11, 38, 53,  26;\n  7: 0, 0, 0, 0,  4, 30, 82,  97,  42;\n  8: 0, 0, 0, 0,  1, 16, 79, 173, 176,  68;\n  9: 0, 0, 0, 0,  0,  5, 50, 191, 352, 315, 110;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (x+2*x^2+x^3)*P(n-2,x) + (x+x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1 + 2*x, P(1,x) = x + 2*x^2.","mathematica":"P[0, x_] := 1 + 2*x;\nP[1, x_] := x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((x + 2*x^2 + x^3)*P[n - 2, x] + (x + x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = ((x + 2*x^2 + x^3)*v[i - 2] + (x + x^2)*v[i - 1])/(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-f239c3b0ba932394","steps":["x \u21a6 x/(x+1)"],"edges":["e-68f8f8a28cba92c4"],"text":"A199881 \u2192 x \u21a6 x/(x+1)"},"headline":"A199881 \u2192 x \u21a6 x/(x+1)","chain":null}