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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) = 1 + (4) t + (4) 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":"6e0c147a8a1da9cae911a7c53a59d291cfe5687c4c25cf17f6aa135ef5f7810d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-4a68513ed3177e7f","kind":"oeis_match","node_a_id":"n-65e47d0258ed4170","node_b_id":null,"fingerprint":null,"oeis_id":"A008958","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":false,"oeis_id":"A008958","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6761e7e8ff053987","kind":"oeis_match","node_a_id":"n-65e47d0258ed4170","node_b_id":null,"fingerprint":null,"oeis_id":"A160562","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":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A160562","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A160562","relation":"identification"},{"oeis_id":"A008958","relation":"match_reversed_triangle"},{"oeis_id":"A160562","relation":"match_triangle"}],"bfiles":[{"node_id":"n-65e47d0258ed4170","oeis_id":"A008958","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":40209},{"node_id":"n-65e47d0258ed4170","oeis_id":"A160562","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":40209}],"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,        1;\n  2: 1,       10,          1;\n  3: 1,       91,         35,           1;\n  4: 1,      820,        966,          84,           1;\n  5: 1,     7381,      24970,        5082,         165,          1;\n  6: 1,    66430,     631631,      273988,       18447,        286,        1;\n  7: 1,   597871,   15857205,    14057043,     1768195,      53053,      455,      1;\n  8: 1,  5380840,  397027996,   704652312,   157280838,    8187608,   129948,    680,   1;\n  9: 1, 48427561, 9931080740, 34924991284, 13444400190, 1147981406, 30148820, 282948, 969, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*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 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A008958","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"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: 1,  10,      1;\n  3: 1,  35,     91,        1;\n  4: 1,  84,    966,      820,          1;\n  5: 1, 165,   5082,    24970,       7381,           1;\n  6: 1, 286,  18447,   273988,     631631,       66430,           1;\n  7: 1, 455,  53053,  1768195,   14057043,    15857205,      597871,          1;\n  8: 1, 680, 129948,  8187608,  157280838,   704652312,   397027996,    5380840,        1;\n  9: 1, 969, 282948, 30148820, 1147981406, 13444400190, 34924991284, 9931080740, 48427561, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];\nTable[Reverse[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 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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 = Vecrev(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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A160562","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,        1;\n  2: 1,       10,          1;\n  3: 1,       91,         35,           1;\n  4: 1,      820,        966,          84,           1;\n  5: 1,     7381,      24970,        5082,         165,          1;\n  6: 1,    66430,     631631,      273988,       18447,        286,        1;\n  7: 1,   597871,   15857205,    14057043,     1768195,      53053,      455,      1;\n  8: 1,  5380840,  397027996,   704652312,   157280838,    8187608,   129948,    680,   1;\n  9: 1, 48427561, 9931080740, 34924991284, 13444400190, 1147981406, 30148820, 282948, 969, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*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] = (1 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-65e47d0258ed4170","steps":[],"edges":[],"text":"A160562"},"headline":"A160562: Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).","chain":null}