{"id":"n-3491b28f433aa954","revision":1,"first_index":0,"fingerprint":"f6-ccfaa2b0b0cc2c4e","name":"reversal of A natural number operator.","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"],["3","2"],["5","4"],["7","6"],["9","8"],["11","10"],["13","12"],["15","14"],["17","16"],["19","18"],["21","20"],["23","22"],["25","24"],["27","26"],["29","28"]],"degrees":[0,1,1,1,1,1,1,1,1,1,1,1,1,1,1],"row_sums":["1","5","9","13","17","21","25","29","33","37","41","45","49","53","57"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160978,"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-407378711040f171"]},{"id":160979,"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-407378711040f171"]},{"id":160980,"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-407378711040f171"]}],"region_exact":{},"incoming_edges":[{"id":"e-407378711040f171","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-3491b28f433aa954","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":"4f2dafaf07e69a9c8593dce6262e9971c61373b443b826aee4058b47217e2366","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-f4c1dee76f5a68b2","role":"sequence","name":"A natural number operator.","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-e0e2daafb52b00a0","kind":"fingerprint_related","node_a_id":"n-f4c1dee76f5a68b2","node_b_id":"n-3491b28f433aa954","fingerprint":"f6-ccfaa2b0b0cc2c4e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ccfaa2b0b0cc2c4e"},"other_id":"n-f4c1dee76f5a68b2","other":{"id":"n-f4c1dee76f5a68b2","name":"A natural number operator.","first_index":0,"fingerprint":"f6-ccfaa2b0b0cc2c4e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["0","4","5"]],"card_rows":[["1"],["2","3"],["0","4","5"],["0","0","6","7"]],"row_sums_preview":["1","5","9","13","17","21","25","29"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A132774"],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["2D+1+2n"],"edges":["e-3fa49e62bb44b5e9"],"text":"A023531 \u2192 2D+1+2n"},"headline":"A132774: A natural number operator.","chain":"A023531 \u2192 2D+1+2n"}},{"id":"l-681dcf85ebb4f683","kind":"oeis_match","node_a_id":"n-3491b28f433aa954","node_b_id":null,"fingerprint":null,"oeis_id":"A016813","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":"A016813","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9daac96a7778965c","kind":"oeis_match","node_a_id":"n-3491b28f433aa954","node_b_id":null,"fingerprint":null,"oeis_id":"A222945","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A222945","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-edc7f4ed88dca76a","kind":"oeis_match","node_a_id":"n-3491b28f433aa954","node_b_id":null,"fingerprint":null,"oeis_id":"A004766","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A004766","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A004766","relation":"match_row_sums"},{"oeis_id":"A016813","relation":"match_row_sums"},{"oeis_id":"A222945","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-3491b28f433aa954","oeis_id":"A004766","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":56,"oeis_terms_available":56,"term_count":200,"raw_size":1466},{"node_id":"n-3491b28f433aa954","oeis_id":"A016813","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":60,"oeis_terms_available":60,"term_count":200,"raw_size":1462}],"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:  3,  2;\n  2:  5,  4;\n  3:  7,  6;\n  4:  9,  8;\n  5: 11, 10;\n  6: 13, 12;\n  7: 15, 14;\n  8: 17, 16;\n  9: 19, 18;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+2*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + 2*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + 2*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*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-6fea5aff4f44fa54","steps":["2D+1+2n","reverse"],"edges":["e-3fa49e62bb44b5e9","e-407378711040f171"],"text":"A023531 \u2192 2D+1+2n \u2192 reverse"},"headline":"A023531 \u2192 2D+1+2n \u2192 reverse","chain":null}
