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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 + (2) t + (1) t^2 and u + v theta = 3 + (2) theta: 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":"18730f2f125ef0debb08e42fed62a40169684ad5fc886cf4274a790e9f636b78","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-1acec24c4ea49dcb","kind":"oeis_match","node_a_id":"n-67b93749ab954f15","node_b_id":null,"fingerprint":null,"oeis_id":"A156289","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":39},"in_catalog":true,"oeis_id":"A156289","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b8b2c5aad8e9804a","kind":"oeis_match","node_a_id":"n-67b93749ab954f15","node_b_id":null,"fingerprint":null,"oeis_id":"A005046","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A005046","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A156289","relation":"identification"},{"oeis_id":"A005046","relation":"match_row_sums"},{"oeis_id":"A156289","relation":"match_triangle"}],"bfiles":[{"node_id":"n-67b93749ab954f15","oeis_id":"A156289","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":39,"oeis_terms_agreeing":39,"oeis_terms_available":39,"term_count":990,"raw_size":50927}],"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,      3;\n   3: 1,     15,        15;\n   4: 1,     63,       210,        105;\n   5: 1,    255,      2205,       3150,         945;\n   6: 1,   1023,     21120,      65835,       51975,        10395;\n   7: 1,   4095,    195195,    1201200,     1891890,       945945,       135135;\n   8: 1,  16383,   1777230,   20585565,    58108050,     54864810,     18918900,     2027025;\n   9: 1,  65535,  16076985,  341809650,  1637971335,   2614321710,   1640268630,   413513100,   34459425;\n  10: 1, 262143, 145020540, 5581493295, 44025570225, 112133266245, 114359345100, 51068867850, 9820936125, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x)*P(n-1,x) + (3*x+2*x^2)*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[(1 + 3*x)*P[n - 1, x] + (3*x + 2*x^2)*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] = (1 + 3*x)*v[i - 1] + (3*x + 2*x^2)*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":"A156289","target":"triangle","first_row":1,"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,      3;\n   3: 1,     15,        15;\n   4: 1,     63,       210,        105;\n   5: 1,    255,      2205,       3150,         945;\n   6: 1,   1023,     21120,      65835,       51975,        10395;\n   7: 1,   4095,    195195,    1201200,     1891890,       945945,       135135;\n   8: 1,  16383,   1777230,   20585565,    58108050,     54864810,     18918900,     2027025;\n   9: 1,  65535,  16076985,  341809650,  1637971335,   2614321710,   1640268630,   413513100,   34459425;\n  10: 1, 262143, 145020540, 5581493295, 44025570225, 112133266245, 114359345100, 51068867850, 9820936125, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x)*P(n-1,x) + (3*x+2*x^2)*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[(1 + 3*x)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*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 + 3*x)*v[i - 1] + (3*x + 2*x^2)*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 - #c)); 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":true}}],"derivation":{"root":"n-67b93749ab954f15","steps":[],"edges":[],"text":"A156289"},"headline":"A156289: Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.","chain":null}