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an exact degree step keeps every row nonzero; S_n = kappa (theta + a)(theta + b) + x (u + v theta) with kappa (t + a)(t + b) = 4 + -4 n + n^2 + (-4 + 2 n) t + (1) t^2 and u + v theta = 1: The degree-local Euler step: T = kappa (theta + a)(theta + b) + x (u + v theta), theta = x D, with kappa, a, b > 0, u + v k > 0 for every k <= d and a negative comparison defect kappa (a b - (a + b + 1)^2/4) + (u - (a + b + 1) v/2) s - v^2 s^2/(4 kappa) < 0 for every s <= 0, maps every polynomial of degree <= d with nonnegative coefficients and only real roots to such a polynomial (RealRooted IsPFPolynomial.generalStep_of_natDegree_le, a half-line theorem: the symbol of T has discriminant -4 kappa s^2 times the comparison defect at s, so its real-rootedness is required for s <= 0 only, and the hypothesis (i) of P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5, for every real s, is not needed); here d = deg P_(n-1), so u(n) + v(n) k > 0 is required for 0 <= k <= deg P_(n-1) only (both ends, the condition being linear in k); by linearity a row with nonpositive coefficients is handled through its negative","lean_route":null,"lean_declarations":["RealRooted.IsPFPolynomial.generalStep_of_natDegree_le","RealRooted.EulerBidiagonal.generalStep","RealRooted.EulerBidiagonal.comparisonDefect","RealRooted.EulerBidiagonal.generalRowsDep_spec"],"certificate_hash":"e08515297ae114f424239cb6f4b60ecebc92ac907d8eddfe99d49a53dec4db79","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-3d569279f3a09daf","kind":"oeis_match","node_a_id":"n-313c8ccf28f71545","node_b_id":null,"fingerprint":null,"oeis_id":"A390433","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A390433","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-448660ee8fc16e33","kind":"oeis_match","node_a_id":"n-313c8ccf28f71545","node_b_id":null,"fingerprint":null,"oeis_id":"A390667","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"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":"A390667","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A390433","relation":"identification"},{"oeis_id":"A390667","relation":"match_row_sums"},{"oeis_id":"A390433","relation":"match_triangle"}],"bfiles":[{"node_id":"n-313c8ccf28f71545","oeis_id":"A390433","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":45787},{"node_id":"n-313c8ccf28f71545","oeis_id":"A390667","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":68365}],"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: 0,            1;\n   3: 0,            4,            1;\n   4: 0,           36,           20,            1;\n   5: 0,          576,          536,           56,           1;\n   6: 0,        14400,        19872,         3280,         120,          1;\n   7: 0,       518400,       988128,       229792,       13000,        220,        1;\n   8: 0,     25401600,     63758592,     19601280,     1529792,      39620,      364,      1;\n   9: 0,   1625702400,   5189847552,   2023886592,   204706112,    7235072,   101136,    560,   1;\n  10: 0, 131681894400, 520610457600, 250080125184, 31501566720, 1427433280, 27057728, 227136, 816, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-4*n+4+x)*P(n-1,x) + (2*n-3)*x*d/dx P(n-1,x) + x^2*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[(n^2 - 4*n + 4 + x)*P[n - 1, x] + (2*n - 3)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n^2 - 4*n + 4 + x)*v[i - 1] + (2*n - 3)*x*deriv(v[i - 1], x) + x^2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A390433","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: 0,            1;\n  2: 0,            4,            1;\n  3: 0,           36,           20,            1;\n  4: 0,          576,          536,           56,           1;\n  5: 0,        14400,        19872,         3280,         120,          1;\n  6: 0,       518400,       988128,       229792,       13000,        220,        1;\n  7: 0,     25401600,     63758592,     19601280,     1529792,      39620,      364,      1;\n  8: 0,   1625702400,   5189847552,   2023886592,   204706112,    7235072,   101136,    560,   1;\n  9: 0, 131681894400, 520610457600, 250080125184, 31501566720, 1427433280, 27057728, 227136, 816, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+1+x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) + x^2*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[(n^2 - 2*n + 1 + x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x] + x^2*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] = (n^2 - 2*n + 1 + x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x) + x^2*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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-313c8ccf28f71545","steps":[],"edges":[],"text":"A390433"},"headline":"A390433: Triangle read by rows, Lah numbers with level 2.","chain":null}