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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":"55c662990ffe1eba4daa994f7a85ecf3f405735c05d2fdb2410f9a000aee5290","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-0b0f0dd4d2c8eb05","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 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:       9,          2;\n  3:      59,         84,           3;\n  4:     361,       1850,         462,           4;\n  5:    2175,      32804,       26811,        2028,           5;\n  6:   13061,     531210,     1095150,      270000,        7935,          6;\n  7:   78379,    8255492,    37835535,    22371400,     2211210,      29112,         7;\n  8:  470289,  125870234,  1195610598,  1460739420,   341689610,   15916452,    102732,      8;\n  9: 2821751, 1902162180, 35931566307, 83268896004, 38984832735, 4318589856, 105173810, 353448, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(12*n-6+(8*n^2-24*n+10)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) - ((18*n-9)*x+(28*n^2-60*n+23)*x^2-(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) - ((4*n-2)*x^2+(8*n^2-16*n+6)*x^3)*d^2/dx^2 P(n-2,x) + (14*n-19+(8*n^2-24*n+18)*x)*P(n-1,x) + ((18*n-27)*x-(4*n-6)*x^2)*d/dx P(n-1,x) + (4*n-6)*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[Together[(-(12*n - 6 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] - ((18*n - 9)*x + (28*n^2 - 60*n + 23)*x^2 - (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] - ((4*n - 2)*x^2 + (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], {x, 2}] + (14*n - 19 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] + ((18*n - 27)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x] + (4*n - 6)*x^2*D[P[n - 1, x], {x, 2}])/(2*n - 3)]];\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] = (-(12*n - 6 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] - ((18*n - 9)*x + (28*n^2 - 60*n + 23)*x^2 - (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) - ((4*n - 2)*x^2 + (8*n^2 - 16*n + 6)*x^3)*deriv(deriv(v[i - 2], x), x) + (14*n - 19 + (8*n^2 - 24*n + 18)*x)*v[i - 1] + ((18*n - 27)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x) + (4*n - 6)*x^2*deriv(deriv(v[i - 1], x), x))/(2*n - 3)); 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-0b0f0dd4d2c8eb05","steps":["D"],"edges":["e-d1a2a549d583a284"],"text":"A166961 \u2192 D"},"headline":"A166961 \u2192 D","chain":null}