{"id":"n-a97072050b882c00","revision":0,"first_index":0,"fingerprint":"f6-4482ebdbb18b2450","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["2"],["4"],["8","4"],["16","8"],["32","24"],["64","48"],["128","128","16"],["256","256","32"],["512","640","160"],["1024","1280","320"],["2048","3072","1152","64"],["4096","6144","2304","128"],["8192","14336","7168","896"]],"degrees":[0,0,0,0,1,1,1,1,2,2,2,2,3,3,3],"row_sums":["1","2","2","4","12","24","56","112","272","544","1312","2624","6336","12672","30592"],"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"],["2"],["4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171982,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-e0770db3463d49fd"]},{"id":171983,"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-e0770db3463d49fd"]},{"id":171984,"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-e0770db3463d49fd"]}],"region_exact":{},"incoming_edges":[{"id":"e-e0770db3463d49fd","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a97072050b882c00","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":"cc9ae7ce938d1066965381d1308766a1e639ddefe999f4739de636a8c62eade9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-e414b9b19e04a193","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","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-0139a535d6e81321","kind":"fingerprint_related","node_a_id":"n-30bdaca9204902d8","node_b_id":"n-a97072050b882c00","fingerprint":"f6-4482ebdbb18b2450","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4482ebdbb18b2450"},"other_id":"n-30bdaca9204902d8","other":{"id":"n-30bdaca9204902d8","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-4482ebdbb18b2450","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["1"]],"row_sums_preview":["1","1","1","1","3","3","7","7"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-c2e81851a0a2b89d","e-6103c76c0177f60b"],"text":"A152198 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A152198 \u2192 Veronese 2 \u2192 x \u21a6 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;\n  2:   2;\n  3:   4;\n  4:   8,   4;\n  5:  16,   8;\n  6:  32,  24;\n  7:  64,  48;\n  8: 128, 128, 16;\n  9: 256, 256, 32;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*x*P(n-4,x) + 4*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 2, P(3,x) = 4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 2;\nP[3, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[4*x*P[n - 4, x] + 4*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 2; v[4] = 4; for(n = 4, N, my(i = n + 1); v[i] = 4*x*v[i - 4] + 4*v[i - 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-b6aca576a689e9f0","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-bba306700a33083a","e-e0770db3463d49fd"],"text":"A274496 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A274496 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}