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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"005ab2d37bb86d629a3205801386d38b6884a8c4d60bb13f78d955d7e076a759","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4f94f81718ae3273","role":"sequence","name":"Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"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:            9,            2;\n   2:           79,           36,            3;\n   3:          765,          530,           90,           4;\n   4:         8311,         7740,         1995,         180,           5;\n   5:       100989,       118202,        40950,        5600,         315,          6;\n   6:      1362439,      1920996,       836703,      153720,       13090,        504,         7;\n   7:     20246445,     33401090,     17533530,     4049724,      463050,      26964,       756,       8;\n   8:    328972471,    621639180,    381995715,   106350300,    15284535,    1194480,     50610,    1080,      9;\n   9:   5805917469,  12364629002,   8706916350,  2838239800,   491673105,   48284082,   2744280,   88440,   1485,   10;\n  10: 110645911399, 262204157556, 208177133703, 77783127840, 15777801380, 1867424328, 133337589, 5761800, 146025, 1980, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(14+6*x+x^2)*P(n-2,x) - (18+13*x+3*x^2)*d/dx P(n-2,x) - (6*x+2*x^2)*d^2/dx^2 P(n-2,x) + (9+2*x)*P(n-1,x) + (6+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 9 + 2*x, P(2,x) = 79 + 36*x + 3*x^2.","mathematica":"P[1, x_] := 9 + 2*x;\nP[2, x_] := 79 + 36*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(14 + 6*x + x^2)*P[n - 2, x] - (18 + 13*x + 3*x^2)*D[P[n - 2, x], x] - (6*x + 2*x^2)*D[P[n - 2, x], {x, 2}] + (9 + 2*x)*P[n - 1, x] + (6 + 3*x)*D[P[n - 1, x], x] + 2*x*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] = 9 + 2*x; v[2] = 79 + 36*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(14 + 6*x + x^2)*v[i - 2] - (18 + 13*x + 3*x^2)*deriv(v[i - 2], x) - (6*x + 2*x^2)*deriv(deriv(v[i - 2], x), x) + (9 + 2*x)*v[i - 1] + (6 + 3*x)*deriv(v[i - 1], x) + 2*x*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}}],"derivation":{"root":"n-4f94f81718ae3273","steps":["D"],"edges":["e-1f6777d5e3bfd5ee"],"text":"A088729 \u2192 D"},"headline":"A088729 \u2192 D","chain":null}