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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 + (3) t^2 and u + v theta = -2 + 3 n + (-3) 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":"4f51cc1280603fe9e359dacaf1d6900d1a815b2db351203d253cb4afa67790dd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-31453e7729b4449e","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-2feacf100ed042e2","conclusion_name":"shift_x_plus_one of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n."},{"id":"e-5413d412f5e0676c","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9de429b059cac5ba","conclusion_name":"one_plus_derivative of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n."},{"id":"e-907caf4394cc8d57","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7cb278ca98b33a0d","conclusion_name":"derivative of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n."},{"id":"e-a5ddde62de67fc97","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-d33cbe5548dd9a88","conclusion_name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n."},{"id":"e-bebe1fbc9538e542","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8c87e3909ebe3753","conclusion_name":"theta_plus_one of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n."},{"id":"e-bfb68eea1887527f","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-aaf5f334c8903a15","conclusion_name":"theta_plus_one_squared of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n."},{"id":"e-edc641caac81f316","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-9bc39e02344a1087","conclusion_name":"reversal of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 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0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-25ac1cc25c118491","steps":["reverse"],"edges":["e-edc641caac81f316"],"text":"A166962 \u2192 reverse"},"headline":"A166962 \u2192 reverse","chain":null}},{"id":"l-f2a8e37e6126e114","kind":"oeis_match","node_a_id":"n-25ac1cc25c118491","node_b_id":null,"fingerprint":null,"oeis_id":"A166962","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":41},"in_catalog":true,"oeis_id":"A166962","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A166962","relation":"identification"},{"oeis_id":"A166962","relation":"match_triangle"}],"bfiles":[{"node_id":"n-25ac1cc25c118491","oeis_id":"A166962","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":41,"oeis_terms_agreeing":41,"oeis_terms_available":41,"term_count":990,"raw_size":49312}],"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,       12,           1;\n  3: 1,      103,          69,            1;\n  4: 1,      834,        2170,          316,            1;\n  5: 1,     6685,       53910,        27830,         1329,           1;\n  6: 1,    53496,     1219015,      1652300,       281195,        5412,          1;\n  7: 1,   427987,    26455251,     81939195,     34800675,     2487917,      21781,         1;\n  8: 1,  3423918,   563692024,   3700851816,   3327253410,   586846782,   20312292,     87300,      1;\n  9: 1, 27391369, 11912858700, 158744221096, 275485100706, 99591585402, 8570002656, 157550844, 349425, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n-2)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*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 + (3*n - 2)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*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 + (3*n - 2)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*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":"A166962","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,        1;\n   3: 1,       12,           1;\n   4: 1,      103,          69,            1;\n   5: 1,      834,        2170,          316,            1;\n   6: 1,     6685,       53910,        27830,         1329,           1;\n   7: 1,    53496,     1219015,      1652300,       281195,        5412,          1;\n   8: 1,   427987,    26455251,     81939195,     34800675,     2487917,      21781,         1;\n   9: 1,  3423918,   563692024,   3700851816,   3327253410,   586846782,   20312292,     87300,      1;\n  10: 1, 27391369, 11912858700, 158744221096, 275485100706, 99591585402, 8570002656, 157550844, 349425, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (7*x-3*x^2)*d/dx P(n-1,x) + 3*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*n - 5)*x)*P[n - 1, x] + (7*x - 3*x^2)*D[P[n - 1, x], x] + 3*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*n - 5)*x)*v[i - 1] + (7*x - 3*x^2)*deriv(v[i - 1], x) + 3*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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-25ac1cc25c118491","steps":[],"edges":[],"text":"A166962"},"headline":"A166962: Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","chain":null}