{"id":"n-9c8414944cbb13fd","revision":0,"first_index":0,"fingerprint":"f6-27d55a3d589e9c6a","name":"mobius_x_over_x_plus_one of A skewed version of triangular array A122075.","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"],["0","2"],["0","1","4"],["0","0","3","8"],["0","0","1","9","16"],["0","0","0","4","23","32"],["0","0","0","1","15","57","64"],["0","0","0","0","5","46","135","128"],["0","0","0","0","1","22","133","313","256"],["0","0","0","0","0","6","78","360","711","512"],["0","0","0","0","0","1","30","255","939","1593","1024"],["0","0","0","0","0","0","7","120","771","2370","3527","2048"],["0","0","0","0","0","0","1","39","435","2220","5841","7737","4096"],["0","0","0","0","0","0","0","8","173","1446","6132","14108","16839","8192"],["0","0","0","0","0","0","0","1","49","686","4536","16413","33527","36409","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","5","11","26","59","137","314","725","1667","3842","8843","20369","46898","108005"],"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"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","1","4","5","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171743,"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-ab7b21ac420ddd05"]},{"id":171744,"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-ab7b21ac420ddd05"]},{"id":171745,"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-ab7b21ac420ddd05"]}],"region_exact":{},"incoming_edges":[{"id":"e-ab7b21ac420ddd05","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-9c8414944cbb13fd","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":"c007869b3e7495d73de008dfbc42fadba280305486b069e8e571df09dd572f2e","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-540fd96c9759fa70","role":"sequence","name":"A skewed version of triangular array A122075.","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":[{"id":"l-9aeb5f8bbce1ac04","kind":"fingerprint_related","node_a_id":"n-9c8414944cbb13fd","node_b_id":"n-97af8fbb64a585df","fingerprint":"f6-27d55a3d589e9c6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27d55a3d589e9c6a"},"other_id":"n-97af8fbb64a585df","other":{"id":"n-97af8fbb64a585df","name":"shift_x_plus_one of Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-27d55a3d589e9c6a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","1"]],"card_rows":[["1"],["2"],["4","1"],["8","3"]],"row_sums_preview":["1","2","5","11","26","59","137","314"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-72cbbb2f152c8f9c","steps":["x \u21a6 x+1"],"edges":["e-50cdf5d4275c77e7"],"text":"A209599 \u2192 x \u21a6 x+1"},"headline":"A209599 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d870b80943ecffad","kind":"fingerprint_related","node_a_id":"n-9c8414944cbb13fd","node_b_id":"n-51a5179166bfef55","fingerprint":"f6-27d55a3d589e9c6a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27d55a3d589e9c6a"},"other_id":"n-51a5179166bfef55","other":{"id":"n-51a5179166bfef55","name":"basis_map_A007318 of Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-27d55a3d589e9c6a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","1"]],"card_rows":[["1"],["2"],["4","1"],["8","3"]],"row_sums_preview":["1","2","5","11","26","59","137","314"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-72cbbb2f152c8f9c","steps":["basis A007318"],"edges":["e-2b79b0b0713fe4e6"],"text":"A209599 \u2192 basis A007318"},"headline":"A209599 \u2192 basis A007318","chain":null}}],"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: 0, 2;\n  2: 0, 1, 4;\n  3: 0, 0, 3, 8;\n  4: 0, 0, 1, 9, 16;\n  5: 0, 0, 0, 4, 23, 32;\n  6: 0, 0, 0, 1, 15, 57,  64;\n  7: 0, 0, 0, 0,  5, 46, 135, 128;\n  8: 0, 0, 0, 0,  1, 22, 133, 313, 256;\n  9: 0, 0, 0, 0,  0,  6,  78, 360, 711, 512;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (x+4*x^2+5*x^3+2*x^4)*P(n-2,x) + (x+2*x^2+x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((x + 4*x^2 + 5*x^3 + 2*x^4)*P[n - 2, x] + (x + 2*x^2 + x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((x + 4*x^2 + 5*x^3 + 2*x^4)*v[i - 2] + (x + 2*x^2 + x^3)*v[i - 1])/(1 + 2*x + x^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-540fd96c9759fa70","steps":["x \u21a6 x/(x+1)"],"edges":["e-ab7b21ac420ddd05"],"text":"A236076 \u2192 x \u21a6 x/(x+1)"},"headline":"A236076 \u2192 x \u21a6 x/(x+1)","chain":null}