{"id":"n-33a4ac1ca078a4fe","revision":0,"first_index":1,"fingerprint":"f6-adf2adbefbd3f634","name":"reversal of shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","4"],["1","9","17"],["1","15","64","77"],["1","22","157","427","372"],["1","30","315","1435","2821","1915"],["1","39","560","3760","12376","18890","10481"],["1","49","917","8435","40936","104426","129683","60814"],["1","60","1414","16982","113001","424718","879047","918081","372939"],["1","72","2082","31542","274323","1413636","4304077","7466517","6721438","2409837"],["1","85","2955","55020","604317","4071795","16984780","43249600","64432771","50950061","16360786"],["1","99","4070","91245","1233177","10500105","57256837","199486815","435119091","567295487","399976633","116393205"],["1","114","5467","145145","2364747","24795012","170890126","777798978","2317297796","4411076384","5109106828","3250867344","865549453"],["1","130","7189","222937","4306302","54478281","462553234","2660265322","10342829354","26838453642","45255832700","47138737464","27338099829","6713065156"],["1","147","9282","332332","7506499","112689577","1155049324","8184177144","40183355212","135885509760","311677608606","471283380932","445938022713","237681039229","54190360453"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","27","157","979","6517","46107","345261","2726243","22623525","196712171","1787356765","16929897395","166808851541","1706299041211"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2","-2"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180997,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-991edd6a63cf1994"]},{"id":180998,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-991edd6a63cf1994"]},{"id":180999,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-991edd6a63cf1994"]}],"region_exact":{},"incoming_edges":[{"id":"e-991edd6a63cf1994","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-33a4ac1ca078a4fe","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":"95bec0dbc2a6aa668eaf05aee62a8fa61b3fc5b7ac56c4bcff7404709ecc7d8d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c13c144ce0d4f57d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-02ab6a7895aca9b7","kind":"candidate_equal","node_a_id":"n-753eebbdd6abb3d8","node_b_id":"n-33a4ac1ca078a4fe","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-753eebbdd6abb3d8","other":{"id":"n-753eebbdd6abb3d8","name":"mobius_x_over_x_plus_one of reversal of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-adf2adbefbd3f634","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","9","17"]],"card_rows":[["1"],["1","4"],["1","9","17"],["1","15","64","77"]],"row_sums_preview":["1","5","27","157","979","6517","46107","345261"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-0c475173369de966","e-c072fa94a843a7aa"],"text":"A143495 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A143495 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-548608be65186821","kind":"fingerprint_related","node_a_id":"n-c13c144ce0d4f57d","node_b_id":"n-33a4ac1ca078a4fe","fingerprint":"f6-adf2adbefbd3f634","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-adf2adbefbd3f634"},"other_id":"n-c13c144ce0d4f57d","other":{"id":"n-c13c144ce0d4f57d","name":"shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-adf2adbefbd3f634","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["17","9","1"]],"card_rows":[["1"],["4","1"],["17","9","1"],["77","64","15","1"]],"row_sums_preview":["1","5","27","157","979","6517","46107","345261"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["x \u21a6 x+1"],"edges":["e-6768e07b6c7820a7"],"text":"A143495 \u2192 x \u21a6 x+1"},"headline":"A143495 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e9c1b36f472ff40e","kind":"fingerprint_related","node_a_id":"n-ab223477daf211c6","node_b_id":"n-33a4ac1ca078a4fe","fingerprint":"f6-adf2adbefbd3f634","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-adf2adbefbd3f634"},"other_id":"n-ab223477daf211c6","other":{"id":"n-ab223477daf211c6","name":"shift_x_plus_one of basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-adf2adbefbd3f634","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["17","9","1"]],"card_rows":[["1"],["4","1"],["17","9","1"],["77","64","15","1"]],"row_sums_preview":["1","5","27","157","979","6517","46107","345261"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bdae5aa3d1cac0ec","steps":["D+1","D+1"],"edges":["e-70027a74f76c6c1d","e-7fccaa8f4c74fc62"],"text":"A137597 \u2192 D+1 \u2192 D+1"},"headline":"A137597 \u2192 D+1 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  4;\n   3: 1,  9,   17;\n   4: 1, 15,   64,    77;\n   5: 1, 22,  157,   427,    372;\n   6: 1, 30,  315,  1435,   2821,    1915;\n   7: 1, 39,  560,  3760,  12376,   18890,   10481;\n   8: 1, 49,  917,  8435,  40936,  104426,  129683,   60814;\n   9: 1, 60, 1414, 16982, 113001,  424718,  879047,  918081,  372939;\n  10: 1, 72, 2082, 31542, 274323, 1413636, 4304077, 7466517, 6721438, 2409837;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n+2)*x+(n-2)*x^2)*P(n-1,x) - (x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n + 2)*x + (n - 2)*x^2)*P[n - 1, x] - (x^2 + x^3)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n + 2)*x + (n - 2)*x^2)*v[i - 1] - (x^2 + x^3)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["x \u21a6 x+1","reverse"],"edges":["e-6768e07b6c7820a7","e-991edd6a63cf1994"],"text":"A143495 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A143495 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}
