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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":"5c47112a746a3f217881bb09abcd5eff3714d914c3bfd4d25094acc52f2293da","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b8808e9caf8dcf35","role":"sequence","name":"Triangle read by rows, (2, 3)-Lah numbers.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:                  0;\n  1:                  1;\n  2:                100,                 2;\n  3:               9864,               400,                3;\n  4:            1099296,             68128,             1032,              4;\n  5:          142159392,          12009024,           276600,           2160,            5;\n  6:        21385410048,        2313843840,         72227136,         854800,         3980,          6;\n  7:      3724783667712,      496284212736,      19721871360,      315131648,      2218720,       6720,        7;\n  8:    745991108093952,   119113515201024,    5793225387264,   117368874752,   1112779840,    5088384,    10640,     8;\n  9: 170540762379724800, 31985042107650048, 1852912409720832, 45751815936000, 548424615680, 3370689408, 10628688, 16032, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+18*n^3+121*n^2+360*n+400+(2*n^2+16*n+32)*x+x^2)*P(n-2,x) - ((2*n^3+27*n^2+120*n+176)*x+(2*n+10)*x^2)*d/dx P(n-2,x) - ((n^2+8*n+16)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+20*n+52+2*x)*P(n-1,x) + (2*n+13)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 18*n^3 + 121*n^2 + 360*n + 400 + (2*n^2 + 16*n + 32)*x + x^2)*P[n - 2, x] - ((2*n^3 + 27*n^2 + 120*n + 176)*x + (2*n + 10)*x^2)*D[P[n - 2, x], x] - ((n^2 + 8*n + 16)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 20*n + 52 + 2*x)*P[n - 1, x] + (2*n + 13)*x*D[P[n - 1, x], x] + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 18*n^3 + 121*n^2 + 360*n + 400 + (2*n^2 + 16*n + 32)*x + x^2)*v[i - 2] - ((2*n^3 + 27*n^2 + 120*n + 176)*x + (2*n + 10)*x^2)*deriv(v[i - 2], x) - ((n^2 + 8*n + 16)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 20*n + 52 + 2*x)*v[i - 1] + (2*n + 13)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b8808e9caf8dcf35","steps":["D"],"edges":["e-bd1cbe1621b44760"],"text":"A371259 \u2192 D"},"headline":"A371259 \u2192 D","chain":null}