{"id":"n-6115e40a0e752de9","revision":1,"first_index":1,"fingerprint":"f6-bf3a8403ddb100bf","name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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","1"],["2","4","1"],["2","10","7","1"],["2","18","26","10","1"],["2","26","66","50","13","1"],["2","34","130","166","82","16","1"],["2","42","218","418","338","122","19","1"],["2","50","330","870","1042","602","170","22","1"],["2","58","466","1586","2610","2202","978","226","25","1"],["2","66","626","2630","5618","6474","4146","1486","290","28","1"],["2","74","810","4066","10802","16154","14026","7170","2146","362","31","1"],["2","82","1018","5958","19058","35546","39930","27470","11618","2978","442","34","1"],["2","90","1250","8370","31442","70906","99330","88034","49762","17882","4002","530","37","1"],["2","98","1506","11366","49170","130826","221954","244878","177506","84722","26402","5238","626","40","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","20","57","158","431","1160","3089","8154","21367","55644","144137","371638","954335"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1","1"],["2","4","1"],["2","10","7","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161312,"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-c74a6074373531d8"]},{"id":161313,"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-c74a6074373531d8"]},{"id":161314,"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-c74a6074373531d8"]},{"id":161315,"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-c74a6074373531d8"]}],"region_exact":{},"incoming_edges":[{"id":"e-c74a6074373531d8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-6115e40a0e752de9","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"66aec9f73723b668b9f3c9bfec167997b78e762fef77b4727f7bb7ca1c0df747","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b0fc620949fb5990","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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-7909d788850aab09","kind":"oeis_match","node_a_id":"n-6115e40a0e752de9","node_b_id":null,"fingerprint":null,"oeis_id":"A292400","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":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A292400","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A292400","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-6115e40a0e752de9","oeis_id":"A292400","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":29,"oeis_terms_available":29,"term_count":200,"raw_size":8943}],"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,  1;\n   3: 2,  4,   1;\n   4: 2, 10,   7,    1;\n   5: 2, 18,  26,   10,    1;\n   6: 2, 26,  66,   50,   13,    1;\n   7: 2, 34, 130,  166,   82,   16,   1;\n   8: 2, 42, 218,  418,  338,  122,  19,   1;\n   9: 2, 50, 330,  870, 1042,  602, 170,  22,  1;\n  10: 2, 58, 466, 1586, 2610, 2202, 978, 226, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*x+2*x^2)*P(n-3,x) - (1+x^2)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 2 + 4*x + x^2, P(4,x) = 2 + 10*x + 7*x^2 + x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 2 + 4*x + x^2;\nP[4, x_] := 2 + 10*x + 7*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*x + 2*x^2)*P[n - 3, x] - (1 + x^2)*P[n - 2, x] + (2 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 2 + 4*x + x^2; v[4] = 2 + 10*x + 7*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = -x^2*v[i - 4] - (2*x + 2*x^2)*v[i - 3] - (1 + x^2)*v[i - 2] + (2 + 2*x)*v[i - 1]); 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-b0fc620949fb5990","steps":["D+1"],"edges":["e-c74a6074373531d8"],"text":"A122542 \u2192 D+1"},"headline":"A122542 \u2192 D+1","chain":null}
