{"id":"n-3524478ec50f6156","revision":0,"first_index":0,"fingerprint":"f6-6b183ed9136cbe2a","name":"reversal of theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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":true,"rows":[["1"],["2","1"],["3"],["4","3"],["5"],["6","5"],["7"],["8","7"],["9"],["10","9"],["11"],["12","11"],["13"],["14","13"],["15"]],"degrees":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],"row_sums":["1","3","3","7","5","11","7","15","9","19","11","23","13","27","15"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","0"],["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","0"],["0","0","1"],["0","-2","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171024,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-7a297f88c8d2dc71"]},{"id":171025,"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-7a297f88c8d2dc71"]},{"id":171026,"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-7a297f88c8d2dc71"]}],"region_exact":{},"incoming_edges":[{"id":"e-7a297f88c8d2dc71","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-3524478ec50f6156","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"5a225e9c375eb12616194a94e662ce3503dab482522a8bc1b973cc7bbd3d30d0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9f865b2dcffd9592","role":"sequence","name":"theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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-991d1df9cf5671ff","kind":"fingerprint_related","node_a_id":"n-9f865b2dcffd9592","node_b_id":"n-3524478ec50f6156","fingerprint":"f6-6b183ed9136cbe2a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b183ed9136cbe2a"},"other_id":"n-9f865b2dcffd9592","other":{"id":"n-9f865b2dcffd9592","name":"theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","first_index":0,"fingerprint":"f6-6b183ed9136cbe2a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["0","0","3"]],"card_rows":[["1"],["1","2"],["0","0","3"],["0","0","3","4"]],"row_sums_preview":["1","3","3","7","5","11","7","15"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea2139b8dbf6bf8d","steps":["\u03b8+1"],"edges":["e-246a0576b6d0a23b"],"text":"A133080 \u2192 \u03b8+1"},"headline":"A133080 \u2192 \u03b8+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:  3;\n  3:  4, 3;\n  4:  5;\n  5:  6, 5;\n  6:  7;\n  7:  8, 7;\n  8:  9;\n  9: 10, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n)*P(n,x) = (n^2-1+(n^2-n)*x)*P(n-1,x) - ((2*n^2-1)*x+(n^2-n)*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[((n^2 - 1 + (n^2 - n)*x)*P[n - 1, x] - ((2*n^2 - 1)*x + (n^2 - n)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];\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] = ((n^2 - 1 + (n^2 - n)*x)*v[i - 1] - ((2*n^2 - 1)*x + (n^2 - n)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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-ea2139b8dbf6bf8d","steps":["\u03b8+1","reverse"],"edges":["e-246a0576b6d0a23b","e-7a297f88c8d2dc71"],"text":"A133080 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A133080 \u2192 \u03b8+1 \u2192 reverse","chain":null}