{"id":"n-b99dbe4a320ec92e","revision":1,"first_index":0,"fingerprint":"f6-4240140caa375ad0","name":"theta_plus_one 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":true,"rows":[["1"],["2","6"],["0","8","15"],["0","0","18","28"],["0","0","0","32","45"],["0","0","0","0","50","66"],["0","0","0","0","0","72","91"],["0","0","0","0","0","0","98","120"],["0","0","0","0","0","0","0","128","153"],["0","0","0","0","0","0","0","0","162","190"],["0","0","0","0","0","0","0","0","0","200","231"],["0","0","0","0","0","0","0","0","0","0","242","276"],["0","0","0","0","0","0","0","0","0","0","0","288","325"],["0","0","0","0","0","0","0","0","0","0","0","0","338","378"],["0","0","0","0","0","0","0","0","0","0","0","0","0","392","435"]],"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:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","0"],["-4","-1"],["2","2"]]},"terms":[{"coeff":{"coeffs":[["0","0","1"],["0","0","3"],["0","2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160969,"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-9cac8cec827e3b4d"]},{"id":160970,"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-9cac8cec827e3b4d"]},{"id":160971,"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-9cac8cec827e3b4d"]}],"region_exact":{},"incoming_edges":[{"id":"e-9cac8cec827e3b4d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-b99dbe4a320ec92e","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"e394940c408b769d0b3c81e69b1d3b539f6948cc36795a2140f8d40758a51e3b","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-e02aad6616b85af5","kind":"oeis_match","node_a_id":"n-b99dbe4a320ec92e","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-b99dbe4a320ec92e","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: 2, 6;\n  2: 0, 8, 15;\n  3: 0, 0, 18, 28;\n  4: 0, 0,  0, 32, 45;\n  5: 0, 0,  0,  0, 50, 66;\n  6: 0, 0,  0,  0,  0, 72, 91;\n  7: 0, 0,  0,  0,  0,  0, 98, 120;\n  8: 0, 0,  0,  0,  0,  0,  0, 128, 153;\n  9: 0, 0,  0,  0,  0,  0,  0,   0, 162, 190;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-4*n+2+(2*n^2-n)*x)*P(n,x) = (2*n^2*x+(2*n^2+3*n+1)*x^2)*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*x + (2*n^2 + 3*n + 1)*x^2)*P[n - 1, x])/(2*n^2 - 4*n + 2 + (2*n^2 - n)*x)]];\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*x + (2*n^2 + 3*n + 1)*x^2)*v[i - 1])/(2*n^2 - 4*n + 2 + (2*n^2 - n)*x)); 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","\u03b8+1"],"edges":["e-3fa49e62bb44b5e9","e-9cac8cec827e3b4d"],"text":"A023531 \u2192 2D+1+2n \u2192 \u03b8+1"},"headline":"A023531 \u2192 2D+1+2n \u2192 \u03b8+1","chain":null}