{"id":"n-ae37fdb1a61214fb","revision":1,"first_index":1,"fingerprint":"f6-e61870f548717ba4","name":"T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^n*n!*L(n,1-n,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":true,"rows":[["1"],["-1","1"],["0","-2","1"],["0","0","-3","1"],["0","0","0","-4","1"],["0","0","0","0","-5","1"],["0","0","0","0","0","-6","1"],["0","0","0","0","0","0","-7","1"],["0","0","0","0","0","0","0","-8","1"],["0","0","0","0","0","0","0","0","-9","1"],["0","0","0","0","0","0","0","0","0","-10","1"],["0","0","0","0","0","0","0","0","0","0","-11","1"],["0","0","0","0","0","0","0","0","0","0","0","-12","1"],["0","0","0","0","0","0","0","0","0","0","0","0","-13","1"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-14","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","-1","-2","-3","-4","-5","-6","-7","-8","-9","-10","-11","-12","-13"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A132013","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["-1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160326,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-b403369ae2d1e52b"]}],"region_exact":{},"incoming_edges":[{"id":"e-b403369ae2d1e52b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"0\",\"1\"]},{\"numerator\":[\"-1\"],\"denominator\":[\"0\",\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-ae37fdb1a61214fb","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":"5dc050550a6444e7bed5745bb7eb65228d2c6af18168e3af5399cc5e2cb5ad41","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","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-2b165a30c494c089","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A001489","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"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":"A001489","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-694fc4f853ffa911","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A024000","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A024000","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-6f7a8bb657e19aa3","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A022960","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A022960","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-7813d8aca84295b4","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A022959","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A022959","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-9c2daac862eb88b6","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A022958","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A022958","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-bbba4ea62a2626b4","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A132013","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":13,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":13,"target":"triangle","terms_compared":91},"in_catalog":true,"oeis_id":"A132013","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-dbe5813dad1d196a","kind":"oeis_match","node_a_id":"n-ae37fdb1a61214fb","node_b_id":null,"fingerprint":null,"oeis_id":"A001478","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A001478","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A132013","relation":"identification"},{"oeis_id":"A001478","relation":"match_row_sums"},{"oeis_id":"A001489","relation":"match_row_sums"},{"oeis_id":"A022958","relation":"match_row_sums"},{"oeis_id":"A022959","relation":"match_row_sums"},{"oeis_id":"A022960","relation":"match_row_sums"},{"oeis_id":"A024000","relation":"match_row_sums"},{"oeis_id":"A132013","relation":"match_triangle"}],"bfiles":[{"node_id":"n-ae37fdb1a61214fb","oeis_id":"A001478","target":"row_sums","oeis_offset":1,"first_row":3,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":65,"oeis_terms_available":65,"term_count":200,"raw_size":1584},{"node_id":"n-ae37fdb1a61214fb","oeis_id":"A001489","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":200,"raw_size":1579},{"node_id":"n-ae37fdb1a61214fb","oeis_id":"A024000","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":58,"oeis_terms_available":58,"term_count":200,"raw_size":1576},{"node_id":"n-ae37fdb1a61214fb","oeis_id":"A132013","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":91,"oeis_terms_agreeing":91,"oeis_terms_available":91,"term_count":990,"raw_size":5907}],"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:  0, -2,  1;\n   4:  0,  0, -3,  1;\n   5:  0,  0,  0, -4,  1;\n   6:  0,  0,  0,  0, -5,  1;\n   7:  0,  0,  0,  0,  0, -6,  1;\n   8:  0,  0,  0,  0,  0,  0, -7,  1;\n   9:  0,  0,  0,  0,  0,  0,  0, -8,  1;\n  10:  0,  0,  0,  0,  0,  0,  0,  0, -9, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -1 + x;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - x^2*P[n - 2, 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; for(n = 3, N, my(i = n); v[i] = 2*x*v[i - 1] - x^2*v[i - 2]); 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}},{"oeis_id":"A132013","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1: -1,  1;\n  2:  0, -2,  1;\n  3:  0,  0, -3,  1;\n  4:  0,  0,  0, -4,  1;\n  5:  0,  0,  0,  0, -5,  1;\n  6:  0,  0,  0,  0,  0, -6,  1;\n  7:  0,  0,  0,  0,  0,  0, -7,  1;\n  8:  0,  0,  0,  0,  0,  0,  0, -8,  1;\n  9:  0,  0,  0,  0,  0,  0,  0,  0, -9, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -1 + x;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - x^2*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -1 + x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - x^2*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["(\u22121/n)D\u00b2+(1/n)D"],"edges":["e-b403369ae2d1e52b"],"text":"A023531 \u2192 (\u22121/n)D\u00b2+(1/n)D"},"headline":"A132013: T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^n*n!*L(n,1-n,x).","chain":"A023531 \u2192 (\u22121/n)D\u00b2+(1/n)D"}
