{"id":"n-6cb9b08f268a058b","revision":0,"first_index":1,"fingerprint":"f6-690bd482f7f311e9","name":"Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.","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":false,"rows":[["1"],["-1","1"],["0","-3","1"],["0","3","-6","1"],["0","0","15","-10","1"],["0","0","-15","45","-15","1"],["0","0","0","-105","105","-21","1"],["0","0","0","105","-420","210","-28","1"],["0","0","0","0","945","-1260","378","-36","1"],["0","0","0","0","-945","4725","-3150","630","-45","1"],["0","0","0","0","0","-10395","17325","-6930","990","-55","1"],["0","0","0","0","0","10395","-62370","51975","-13860","1485","-66","1"],["0","0","0","0","0","0","135135","-270270","135135","-25740","2145","-78","1"],["0","0","0","0","0","0","-135135","945945","-945945","315315","-45045","3003","-91","1"],["0","0","0","0","0","0","0","-2027025","4729725","-2837835","675675","-75075","4095","-105","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","-2","-2","6","16","-20","-132","28","1216","936","-12440","-23672","138048","469456"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A104556","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["-1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169309,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[0, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, +inf)","region_exact":null,"edges":["e-0a6ae27a6cba0695"]},{"id":169310,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-0a6ae27a6cba0695"]},{"id":169311,"position":2,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-0a6ae27a6cba0695"]}],"region_exact":{},"incoming_edges":[{"id":"e-0a6ae27a6cba0695","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-6cb9b08f268a058b","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":null,"lean_declarations":["RealRooted.threeTerm_interlaces_of_window"],"certificate_hash":"36a51476a79112e4f766499aa6d4fbfda8599c2a9e0d9121129205373af630b1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-c2f7eae1366ecdb1","kind":"oeis_match","node_a_id":"n-6cb9b08f268a058b","node_b_id":null,"fingerprint":null,"oeis_id":"A104556","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":12,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":12,"target":"triangle","terms_compared":78},"in_catalog":true,"oeis_id":"A104556","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A104556","relation":"identification"},{"oeis_id":"A104556","relation":"match_triangle"}],"bfiles":[{"node_id":"n-6cb9b08f268a058b","oeis_id":"A104556","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":78,"oeis_terms_agreeing":78,"oeis_terms_available":78,"term_count":990,"raw_size":12004}],"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, -3,   1;\n   4:  0,  3,  -6,    1;\n   5:  0,  0,  15,  -10,    1;\n   6:  0,  0, -15,   45,  -15,     1;\n   7:  0,  0,   0, -105,  105,   -21,     1;\n   8:  0,  0,   0,  105, -420,   210,   -28,   1;\n   9:  0,  0,   0,    0,  945, -1260,   378, -36,   1;\n  10:  0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) - (n-1)*x*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[x*P[n - 1, x] - (n - 1)*x*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] = x*v[i - 1] - (n - 1)*x*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":"A104556","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, -3,   1;\n  3:  0,  3,  -6,    1;\n  4:  0,  0,  15,  -10,    1;\n  5:  0,  0, -15,   45,  -15,     1;\n  6:  0,  0,   0, -105,  105,   -21,     1;\n  7:  0,  0,   0,  105, -420,   210,   -28,   1;\n  8:  0,  0,   0,    0,  945, -1260,   378, -36,   1;\n  9:  0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) - n*x*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[x*P[n - 1, x] - n*x*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] = x*v[i - 1] - n*x*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-6cb9b08f268a058b","steps":[],"edges":[],"text":"A104556"},"headline":"A104556: Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.","chain":null}