{"id":"n-106df0aaa5357879","revision":1,"first_index":0,"fingerprint":"f6-429c048c3a049f47","name":"mobius_x_over_x_plus_one of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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","3"],["1","7","8"],["1","12","30","21"],["1","18","73","109","55"],["1","25","145","344","365","144"],["1","33","255","850","1416","1164","377"],["1","42","413","1806","4184","5358","3593","987"],["1","52","630","3458","10415","18422","19139","10835","2584"],["1","63","918","6132","22995","52557","75051","65556","32106","6765"],["1","75","1290","10248","46425","131389","241375","288730","217487","93845","17711"],["1","88","1760","16335","87417","297374","673757","1033370","1062895","703618","271321","46368"],["1","102","2343","25047","155628","622512","1689058","3185142","4189578","3778372","2230608","777432","121393"],["1","117","3055","37180","264550","1223365","3889730","8759200","14131190","16262700","13054880","6954060","2211025","317811"],["1","133","3913","53690","432575","2281565","8359845","22001820","42291060","59576300","60920930","44055740","21377365","6248479","832040"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","16","64","256","1024","4096","16384","65536","262144","1048576","4194304","16777216","67108864","268435456"],"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"],["1","3"]],"recurrence":{"denominator":{"coeffs":[["8","16","8"],["-16","-32","-16"],["8","16","8"]]},"terms":[{"coeff":{"coeffs":[["20","84","154","166","106","30"],["-25","-99","-176","-200","-143","-45"],["5","15","22","34","37","15"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","68","268","516","530","280","60"],["0","-8","-58","-156","-200","-124","-30"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","75","195","270","210","87","15"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163505,"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":false,"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-ba8938d63a27da5b"]},{"id":163506,"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-ba8938d63a27da5b"]},{"id":163507,"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-ba8938d63a27da5b"]}],"region_exact":{},"incoming_edges":[{"id":"e-ba8938d63a27da5b","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-106df0aaa5357879","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":"15f2b52a744db709f9c3937ba5c3b2dc13f8310bf6964731d5b3414bff7907d3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0d893be6799b12ab","kind":"oeis_match","node_a_id":"n-106df0aaa5357879","node_b_id":null,"fingerprint":null,"oeis_id":"A155543","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A155543","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e3f6c8c2cc4a40e6","kind":"oeis_match","node_a_id":"n-106df0aaa5357879","node_b_id":null,"fingerprint":null,"oeis_id":"A347478","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A347478","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fac3f1bd12617770","kind":"oeis_match","node_a_id":"n-106df0aaa5357879","node_b_id":null,"fingerprint":null,"oeis_id":"A000302","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000302","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A000302","relation":"match_row_sums"},{"oeis_id":"A155543","relation":"match_row_sums"},{"oeis_id":"A347478","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-106df0aaa5357879","oeis_id":"A000302","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":12972}],"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,  3;\n  2: 1,  7,   8;\n  3: 1, 12,  30,   21;\n  4: 1, 18,  73,  109,    55;\n  5: 1, 25, 145,  344,   365,   144;\n  6: 1, 33, 255,  850,  1416,  1164,   377;\n  7: 1, 42, 413, 1806,  4184,  5358,  3593,   987;\n  8: 1, 52, 630, 3458, 10415, 18422, 19139, 10835,  2584;\n  9: 1, 63, 918, 6132, 22995, 52557, 75051, 65556, 32106, 6765;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-16*n+8+(16*n^2-32*n+16)*x+(8*n^2-16*n+8)*x^2)*P(n,x) = (5*n^2-25*n+20+(15*n^2-99*n+84)*x+(22*n^2-176*n+154)*x^2+(34*n^2-200*n+166)*x^3+(37*n^2-143*n+106)*x^4+(15*n^2-45*n+30)*x^5)*P(n-1,x) + (6-(8*n-68)*x-(58*n-268)*x^2-(156*n-516)*x^3-(200*n-530)*x^4-(124*n-280)*x^5-(30*n-60)*x^6)*d/dx P(n-1,x) + (12*x+75*x^2+195*x^3+270*x^4+210*x^5+87*x^6+15*x^7)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 25*n + 20 + (15*n^2 - 99*n + 84)*x + (22*n^2 - 176*n + 154)*x^2 + (34*n^2 - 200*n + 166)*x^3 + (37*n^2 - 143*n + 106)*x^4 + (15*n^2 - 45*n + 30)*x^5)*P[n - 1, x] + (6 - (8*n - 68)*x - (58*n - 268)*x^2 - (156*n - 516)*x^3 - (200*n - 530)*x^4 - (124*n - 280)*x^5 - (30*n - 60)*x^6)*D[P[n - 1, x], x] + (12*x + 75*x^2 + 195*x^3 + 270*x^4 + 210*x^5 + 87*x^6 + 15*x^7)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8 + (16*n^2 - 32*n + 16)*x + (8*n^2 - 16*n + 8)*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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n^2 - 25*n + 20 + (15*n^2 - 99*n + 84)*x + (22*n^2 - 176*n + 154)*x^2 + (34*n^2 - 200*n + 166)*x^3 + (37*n^2 - 143*n + 106)*x^4 + (15*n^2 - 45*n + 30)*x^5)*v[i - 1] + (6 - (8*n - 68)*x - (58*n - 268)*x^2 - (156*n - 516)*x^3 - (200*n - 530)*x^4 - (124*n - 280)*x^5 - (30*n - 60)*x^6)*deriv(v[i - 1], x) + (12*x + 75*x^2 + 195*x^3 + 270*x^4 + 210*x^5 + 87*x^6 + 15*x^7)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8 + (16*n^2 - 32*n + 16)*x + (8*n^2 - 16*n + 8)*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-353fb5a029e4e669","steps":["x \u21a6 x/(x+1)"],"edges":["e-ba8938d63a27da5b"],"text":"A207607 \u2192 x \u21a6 x/(x+1)"},"headline":"A207607 \u2192 x \u21a6 x/(x+1)","chain":null}
