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real-rootedness)","region_exact":null,"edges":["e-95b52c7a85b14ad7"]},{"id":169993,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"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 >= 1","region_exact":null,"edges":["e-95b52c7a85b14ad7"]}],"region_exact":{},"incoming_edges":[{"id":"e-95b52c7a85b14ad7","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-3338feb205e3b77f","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 1 + 2 x, B = 2 + 12 x + 10 x^2, C = 4 x + 8 x^2 + 4 x^3: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; a linear operator preserving real-rootedness preserves the alternation of two polynomials (P. Branden, Trans. Amer. Math. Soc. 358 (2006), Corollary 2.5)","lean_route":null,"lean_declarations":[],"certificate_hash":"b29ceaa8e58ce19ba72e22489a3058883d3bf860fb9a0ea50163742f2b14fdc4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-15642b481efc4c88","kind":"oeis_match","node_a_id":"n-3338feb205e3b77f","node_b_id":null,"fingerprint":null,"oeis_id":"A000281","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":14},"in_catalog":false,"oeis_id":"A000281","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-384aedc79dd1603f","kind":"oeis_match","node_a_id":"n-3338feb205e3b77f","node_b_id":null,"fingerprint":null,"oeis_id":"A151775","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":7,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":7,"target":"triangle","terms_compared":30},"in_catalog":true,"oeis_id":"A151775","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A151775","relation":"identification"},{"oeis_id":"A000281","relation":"match_row_sums"},{"oeis_id":"A151775","relation":"match_triangle"}],"bfiles":[{"node_id":"n-3338feb205e3b77f","oeis_id":"A000281","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":14,"oeis_terms_available":14,"term_count":200,"raw_size":83045},{"node_id":"n-3338feb205e3b77f","oeis_id":"A151775","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":30,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":990,"raw_size":85141}],"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:             1,               2;\n  2:             5,              28,               24;\n  3:            61,             662,             1320,               720;\n  4:          1385,           24568,            83664,            100800,             40320;\n  5:         50521,         1326122,          6749040,          13335840,          11491200,            3628800;\n  6:       2702765,        98329108,        692699304,        1979524800,        2739623040,         1836172800,          479001600;\n  7:     199360981,      9596075582,      88966364760,      342459477360,      673354281600,       715588473600,       392302310400,       87178291200;\n  8:   19391512145,   1192744081648,   14036558182944,    69485886288000,   181217064188160,    270530842982400,    233114750668800,   108101081088000,    20922789888000;\n  9: 2404879675441, 183983154281042, 2674388403120480, 16475552674505280, 54515459807827200, 106867999164326400, 128349384991027200, 92958909332889600, 37347179950080000, 6402373705728000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + (2+12*x+10*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^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[(1 + 2*x)*P[n - 1, x] + (2 + 12*x + 10*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {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] = (1 + 2*x)*v[i - 1] + (2 + 12*x + 10*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), 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}},{"oeis_id":"A151775","target":"triangle","first_row":0,"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,               2;\n  2:             5,              28,               24;\n  3:            61,             662,             1320,               720;\n  4:          1385,           24568,            83664,            100800,             40320;\n  5:         50521,         1326122,          6749040,          13335840,          11491200,            3628800;\n  6:       2702765,        98329108,        692699304,        1979524800,        2739623040,         1836172800,          479001600;\n  7:     199360981,      9596075582,      88966364760,      342459477360,      673354281600,       715588473600,       392302310400,       87178291200;\n  8:   19391512145,   1192744081648,   14036558182944,    69485886288000,   181217064188160,    270530842982400,    233114750668800,   108101081088000,    20922789888000;\n  9: 2404879675441, 183983154281042, 2674388403120480, 16475552674505280, 54515459807827200, 106867999164326400, 128349384991027200, 92958909332889600, 37347179950080000, 6402373705728000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + (2+12*x+10*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^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[(1 + 2*x)*P[n - 1, x] + (2 + 12*x + 10*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + 2*x)*v[i - 1] + (2 + 12*x + 10*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x)); 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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-3338feb205e3b77f","steps":[],"edges":[],"text":"A151775"},"headline":"A151775: Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.","chain":null}