{"id":"n-817c8f72f1974479","revision":1,"first_index":0,"fingerprint":"f6-191e5ee8809e6c17","name":"one_plus_derivative of A natural number operator.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5","3"],["4","14","5"],["0","12","27","7"],["0","0","24","44","9"],["0","0","0","40","65","11"],["0","0","0","0","60","90","13"],["0","0","0","0","0","84","119","15"],["0","0","0","0","0","0","112","152","17"],["0","0","0","0","0","0","0","144","189","19"],["0","0","0","0","0","0","0","0","180","230","21"],["0","0","0","0","0","0","0","0","0","220","275","23"],["0","0","0","0","0","0","0","0","0","0","264","324","25"],["0","0","0","0","0","0","0","0","0","0","0","312","377","27"],["0","0","0","0","0","0","0","0","0","0","0","0","364","434","29"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","23","46","77","116","163","218","281","352","431","518","613","716","827"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["4","-1","-1"],["-6","-1","2"],["2","2","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","1"],["0","-2","3","2"],["0","2","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160966,"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-c42e136a1c0028cd"]},{"id":160967,"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-c42e136a1c0028cd"]},{"id":160968,"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-c42e136a1c0028cd"]}],"region_exact":{},"incoming_edges":[{"id":"e-c42e136a1c0028cd","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-817c8f72f1974479","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":"b66dbfd687c14a9894e6a918617c1868af4cdf35eba2710d18295ca559275b39","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-053a63e1ec7cd4eb","kind":"oeis_match","node_a_id":"n-817c8f72f1974479","node_b_id":null,"fingerprint":null,"oeis_id":"A033951","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":"A033951","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A033951","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-817c8f72f1974479","oeis_id":"A033951","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":48,"oeis_terms_available":48,"term_count":200,"raw_size":1859}],"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: 5,  3;\n  2: 4, 14,  5;\n  3: 0, 12, 27,  7;\n  4: 0,  0, 24, 44,  9;\n  5: 0,  0,  0, 40, 65, 11;\n  6: 0,  0,  0,  0, 60, 90,  13;\n  7: 0,  0,  0,  0,  0, 84, 119,  15;\n  8: 0,  0,  0,  0,  0,  0, 112, 152,  17;\n  9: 0,  0,  0,  0,  0,  0,   0, 144, 189, 19;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+4+(2*n^2-n-1)*x+(2*n-1)*x^2)*P(n,x) = ((2*n^2-2*n)*x+(2*n^2+3*n)*x^2+(2*n+1)*x^3)*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^2 - 2*n)*x + (2*n^2 + 3*n)*x^2 + (2*n + 1)*x^3)*P[n - 1, x])/(2*n^2 - 6*n + 4 + (2*n^2 - n - 1)*x + (2*n - 1)*x^2)]];\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^2 - 2*n)*x + (2*n^2 + 3*n)*x^2 + (2*n + 1)*x^3)*v[i - 1])/(2*n^2 - 6*n + 4 + (2*n^2 - n - 1)*x + (2*n - 1)*x^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-6fea5aff4f44fa54","steps":["2D+1+2n","D+1"],"edges":["e-3fa49e62bb44b5e9","e-c42e136a1c0028cd"],"text":"A023531 \u2192 2D+1+2n \u2192 D+1"},"headline":"A023531 \u2192 2D+1+2n \u2192 D+1","chain":null}