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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":"748ae416d9fcb5a46ff73f34e77378c8057bb93dcc912ceaf11e63ed8f971eb6","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-df88941ef8650983","role":"sequence","name":"one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 2).","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:       8,        2;\n  3:      52,       30,        3;\n  4:     320,      320,       72,        4;\n  5:    1936,     3000,     1140,      140,       5;\n  6:   11648,    26432,    15120,     3080,     240,      6;\n  7:   69952,   225120,   182448,    54880,    7000,    378,     7;\n  8:  419840,  1879040,  2080512,   868224,  161280,  14112,   560,   8;\n  9: 2519296, 15484800, 22877760, 12747840, 3237360, 408240, 26040, 792, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(96-32*x-12*x^2+x^4)*P(n-4,x) - (32*x-16*x^2-4*x^3+2*x^4)*d/dx P(n-4,x) + (136-24*x-10*x^2)*P(n-3,x) + (40*x-12*x^2-2*x^3)*d/dx P(n-3,x) - (68-4*x-2*x^2)*P(n-2,x) - (16*x-2*x^2)*d/dx P(n-2,x) + 14*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 52 + 30*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 8 + 2*x;\nP[3, x_] := 52 + 30*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(96 - 32*x - 12*x^2 + x^4)*P[n - 4, x] - (32*x - 16*x^2 - 4*x^3 + 2*x^4)*D[P[n - 4, x], x] + (136 - 24*x - 10*x^2)*P[n - 3, x] + (40*x - 12*x^2 - 2*x^3)*D[P[n - 3, x], x] - (68 - 4*x - 2*x^2)*P[n - 2, x] - (16*x - 2*x^2)*D[P[n - 2, x], x] + 14*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\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] = 8 + 2*x; v[4] = 52 + 30*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(96 - 32*x - 12*x^2 + x^4)*v[i - 4] - (32*x - 16*x^2 - 4*x^3 + 2*x^4)*deriv(v[i - 4], x) + (136 - 24*x - 10*x^2)*v[i - 3] + (40*x - 12*x^2 - 2*x^3)*deriv(v[i - 3], x) - (68 - 4*x - 2*x^2)*v[i - 2] - (16*x - 2*x^2)*deriv(v[i - 2], x) + 14*v[i - 1] + 2*x*deriv(v[i - 1], 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-a7e9ebb9860a9dbe","steps":["D+1","D"],"edges":["e-9fe96d700c9448ac","e-fd0a3f7264273615"],"text":"A075497 \u2192 D+1 \u2192 D"},"headline":"A075497 \u2192 D+1 \u2192 D","chain":null}