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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":"dd2fd0d9ef8678aa2a0df1988b4aa8b4f64233cc3a3b94e19510802398030101","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-7b6a59413dc1b477","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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:    1;\n  1:    1;\n  2:    2,      1;\n  3:    6,      7,       1;\n  4:   19,     48,      18,       1;\n  5:   59,    316,     240,      41,       1;\n  6:  180,   1933,    2815,     978,      88,      1;\n  7:  544,  11103,   29370,   18770,    3570,    183,     1;\n  8: 1637,  60952,  277488,  307750,  106295,  12204,   374,   1;\n  9: 4917, 324400, 2428396, 4434678, 2595425, 543824, 40020, 757, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-4+2*x)*P(n,x) = -(3*n-9+(2*n^2-14*n+30)*x+(n^3-10*n^2+36*n-49)*x^2+(3*n^2-19*n+28)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((2*n-6)*x+(2*n^2-15*n+31)*x^2-(n^2-13*n+36)*x^3-(3*n-15)*x^4-2*x^5)*d/dx P(n-2,x) + (4*n-15+(2*n^2-14*n+32)*x+(5*n-17)*x^2+2*x^3)*P(n-1,x) + ((2*n-8)*x-(n-8)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 9 + (2*n^2 - 14*n + 30)*x + (n^3 - 10*n^2 + 36*n - 49)*x^2 + (3*n^2 - 19*n + 28)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((2*n - 6)*x + (2*n^2 - 15*n + 31)*x^2 - (n^2 - 13*n + 36)*x^3 - (3*n - 15)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (4*n - 15 + (2*n^2 - 14*n + 32)*x + (5*n - 17)*x^2 + 2*x^3)*P[n - 1, x] + ((2*n - 8)*x - (n - 8)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 4 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 9 + (2*n^2 - 14*n + 30)*x + (n^3 - 10*n^2 + 36*n - 49)*x^2 + (3*n^2 - 19*n + 28)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((2*n - 6)*x + (2*n^2 - 15*n + 31)*x^2 - (n^2 - 13*n + 36)*x^3 - (3*n - 15)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (4*n - 15 + (2*n^2 - 14*n + 32)*x + (5*n - 17)*x^2 + 2*x^3)*v[i - 1] + ((2*n - 8)*x - (n - 8)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 4 + 2*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-7b6a59413dc1b477","steps":["D+1"],"edges":["e-ff253c56ca542933"],"text":"A102365 \u2192 D+1"},"headline":"A102365 \u2192 D+1","chain":null}