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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"2f24a82117c3fc783c507c1d0c5af51a5be9736224e98bb887674ef5afd46935","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a0dd3d6ef57da7a5","role":"sequence","name":"Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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:       20,          2;\n  3:      159,        318,            3;\n  4:     1138,       8508,         3414,            4;\n  5:     7997,     155756,       233634,        31988,            5;\n  6:    56016,    2438334,      8954664,      4876668,       280080,            6;\n  7:   392155,   35267298,    262977945,    350637260,     88168245,      2352930,           7;\n  8:  2745134,  488166536,   6658455318,  17603500312,  11097425530,   1464499608,    19215938,         8;\n  9: 19215993, 6586952352, 153921299472, 717407484504, 896759355630, 307842598944, 23054333232, 153727944, 9;\n","recurrence_sentence":"The row polynomials satisfy (6*n-5)*P(n,x) = -(216*n+36)*x^2*P(n-4,x) - ((72*n+12)*x+(432*n^2-1152*n+12)*x^2)*P(n-3,x) - ((216*n+36)*x^2-(216*n+36)*x^3)*d/dx P(n-3,x) - (42*n+7+(72*n^2-264*n+26)*x+(216*n^3-972*n^2+1170*n-425)*x^2)*P(n-2,x) - ((36*n+6)*x+(216*n^2-396*n+144)*x^2-(216*n^2-360*n+150)*x^3)*d/dx P(n-2,x) + (48*n-34+(72*n^2-192*n+110)*x)*P(n-1,x) + ((36*n-30)*x-(36*n-30)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 20 + 2*x, P(3,x) = 159 + 318*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 20 + 2*x;\nP[3, x_] := 159 + 318*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(216*n + 36)*x^2*P[n - 4, x] - ((72*n + 12)*x + (432*n^2 - 1152*n + 12)*x^2)*P[n - 3, x] - ((216*n + 36)*x^2 - (216*n + 36)*x^3)*D[P[n - 3, x], x] - (42*n + 7 + (72*n^2 - 264*n + 26)*x + (216*n^3 - 972*n^2 + 1170*n - 425)*x^2)*P[n - 2, x] - ((36*n + 6)*x + (216*n^2 - 396*n + 144)*x^2 - (216*n^2 - 360*n + 150)*x^3)*D[P[n - 2, x], x] + (48*n - 34 + (72*n^2 - 192*n + 110)*x)*P[n - 1, x] + ((36*n - 30)*x - (36*n - 30)*x^2)*D[P[n - 1, x], x])/(6*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 20 + 2*x; v[4] = 159 + 318*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(216*n + 36)*x^2*v[i - 4] - ((72*n + 12)*x + (432*n^2 - 1152*n + 12)*x^2)*v[i - 3] - ((216*n + 36)*x^2 - (216*n + 36)*x^3)*deriv(v[i - 3], x) - (42*n + 7 + (72*n^2 - 264*n + 26)*x + (216*n^3 - 972*n^2 + 1170*n - 425)*x^2)*v[i - 2] - ((36*n + 6)*x + (216*n^2 - 396*n + 144)*x^2 - (216*n^2 - 360*n + 150)*x^3)*deriv(v[i - 2], x) + (48*n - 34 + (72*n^2 - 192*n + 110)*x)*v[i - 1] + ((36*n - 30)*x - (36*n - 30)*x^2)*deriv(v[i - 1], x))/(6*n - 5)); 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-a0dd3d6ef57da7a5","steps":["D"],"edges":["e-72e987cadba8b65b"],"text":"A144443 \u2192 D"},"headline":"A144443 \u2192 D","chain":null}