{"id":"n-e77cd13e9e0a7e61","revision":0,"first_index":0,"fingerprint":"f6-3661b733ab1f9507","name":"shift_x_plus_one of shift_x_plus_one of chebyshev_t","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"],["2","1"],["7","8","2"],["26","45","24","4"],["97","224","184","64","8"],["362","1045","1160","620","160","16"],["1351","4680","6546","4736","1872","384","32"],["5042","20377","34384","31416","16800","5264","896","64"],["18817","86912","171744","189696","128160","54272","14080","2048","128"],["70226","364905","826416","1070856","875232","468144","163968","36288","4608","256"],["262087","1513160","3863730","5746560","5514480","3568768","1578080","471040","90880","10240","512"],["978122","6211909","17654824","29642844","32675808","24835888","13285888","5004032","1300992","222464","22528","1024"],["3650401","25290720","79179384","148134464","184474440","161126400","101237248","46116864","15121152","3481600","534528","49152","2048"],["13623482","102251773","349644152","721253780","1001490880","988618592","713915904","381937920","151417344","43946240","9078784","1264640","106496","4096"],["50843527","410963336","1523900770","3436168960","5263996640","5796329728","4731663552","2909466624","1354424064","475138048","123673088","23166976","2953216","229376","8192"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","99","577","3363","19601","114243","665857","3880899","22619537","131836323","768398401","4478554083","26102926097"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174570,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-3, -1)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, -1)","region_exact":null,"edges":["e-bfb7184a4a2ca735"]},{"id":174571,"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-bfb7184a4a2ca735"]},{"id":174572,"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-bfb7184a4a2ca735"]},{"id":174573,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-bfb7184a4a2ca735"]}],"region_exact":{},"incoming_edges":[{"id":"e-bfb7184a4a2ca735","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e77cd13e9e0a7e61","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":"1d6ac57552782d89900633dad9ba2cecd379bf2616d37727b3b601cd4bea1392","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5fc5274bb6446a33","role":"sequence","name":"shift_x_plus_one of chebyshev_t","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-abb6de019ca38bde","kind":"fingerprint_related","node_a_id":"n-e77cd13e9e0a7e61","node_b_id":"n-3aa1e02b394f91f2","fingerprint":"f6-3661b733ab1f9507","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3661b733ab1f9507"},"other_id":"n-3aa1e02b394f91f2","other":{"id":"n-3aa1e02b394f91f2","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-3661b733ab1f9507","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["8","32","28"]],"card_rows":[["1"],["2","4"],["8","32","28"],["32","192","360","208"]],"row_sums_preview":["1","6","68","792","9232","107616","1254464","14623104"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-55a1228129c4451e","e-ced7a5b47dd933cf"],"text":"A086645 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A086645 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","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:     2,      1;\n  2:     7,      8,      2;\n  3:    26,     45,     24,       4;\n  4:    97,    224,    184,      64,      8;\n  5:   362,   1045,   1160,     620,    160,     16;\n  6:  1351,   4680,   6546,    4736,   1872,    384,     32;\n  7:  5042,  20377,  34384,   31416,  16800,   5264,    896,    64;\n  8: 18817,  86912, 171744,  189696, 128160,  54272,  14080,  2048,  128;\n  9: 70226, 364905, 826416, 1070856, 875232, 468144, 163968, 36288, 4608, 256;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) + (3+4*x+x^2)*d/dx 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[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (3 + 4*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];\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] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (3 + 4*x + x^2)*deriv(v[i - 1], x))/(n - 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-9ae8d431f7b33f22","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-298ee5f2391f30e9","e-bfb7184a4a2ca735"],"text":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
