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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 + 6 x, B = 3 x + 9 x^2, C = x^2 + 2 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":"dc35a0740d6c6050a3cfa1a24ec1ffe93cd9fabd6b0c1c53a3cf077a0ffcc641","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-54082f6304e41b15","kind":"oeis_match","node_a_id":"n-4ef34abfaa61f0d6","node_b_id":null,"fingerprint":null,"oeis_id":"A241171","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A241171","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e4ebad80c4e46902","kind":"oeis_match","node_a_id":"n-4ef34abfaa61f0d6","node_b_id":null,"fingerprint":null,"oeis_id":"A094088","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":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A094088","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A241171","relation":"identification"},{"oeis_id":"A094088","relation":"match_row_sums"},{"oeis_id":"A241171","relation":"match_triangle"}],"bfiles":[{"node_id":"n-4ef34abfaa61f0d6","oeis_id":"A241171","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":64854}],"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,      6;\n  2: 1,     30,        90;\n  3: 1,    126,      1260,         2520;\n  4: 1,    510,     13230,        75600,        113400;\n  5: 1,   2046,    126720,      1580040,       6237000,        7484400;\n  6: 1,   8190,   1171170,     28828800,     227026800,      681080400,       681080400;\n  7: 1,  32766,  10663380,    494053560,    6972966000,    39502663200,     95351256000,      81729648000;\n  8: 1, 131070,  96461910,   8203431600,  196556560200,  1882311631200,   8266953895200,   16672848192000,   12504636144000;\n  9: 1, 524286, 870123240, 133955839080, 5283068427000, 80735951696400, 576371099304000, 2059096751712000, 3563821301040000, 2375880867360000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) + (3*x+9*x^2)*d/dx P(n-1,x) + (x^2+2*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 + 6*x)*P[n - 1, x] + (3*x + 9*x^2)*D[P[n - 1, x], x] + (x^2 + 2*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 + 6*x)*v[i - 1] + (3*x + 9*x^2)*deriv(v[i - 1], x) + (x^2 + 2*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":"A241171","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      6;\n   3: 1,     30,        90;\n   4: 1,    126,      1260,         2520;\n   5: 1,    510,     13230,        75600,        113400;\n   6: 1,   2046,    126720,      1580040,       6237000,        7484400;\n   7: 1,   8190,   1171170,     28828800,     227026800,      681080400,       681080400;\n   8: 1,  32766,  10663380,    494053560,    6972966000,    39502663200,     95351256000,      81729648000;\n   9: 1, 131070,  96461910,   8203431600,  196556560200,  1882311631200,   8266953895200,   16672848192000,   12504636144000;\n  10: 1, 524286, 870123240, 133955839080, 5283068427000, 80735951696400, 576371099304000, 2059096751712000, 3563821301040000, 2375880867360000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) + (3*x+9*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 6*x)*P[n - 1, x] + (3*x + 9*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + 6*x)*v[i - 1] + (3*x + 9*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-4ef34abfaa61f0d6","steps":[],"edges":[],"text":"A241171"},"headline":"A241171: Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.","chain":null}