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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":"2b4df347f8cb3efabc35eb495f36ad2d27c1aba11c92ca419651ea4c8cb7bdd7","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-5426b03c17935143","role":"sequence","name":"reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] 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:    0;\n  1:    1;\n  2:    4,    2;\n  3:   15,   14,     3;\n  4:   24,   44,    24,     4;\n  5:   81,  180,   138,    44,     5;\n  6:  108,  342,   408,   228,    60,    6;\n  7:  351, 1242,  1737,  1228,   465,   90,    7;\n  8:  432, 1944,  3600,  3544,  2000,  648,  112,   8;\n  9: 1377, 6696, 13716, 15432, 10430, 4344, 1092, 152, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-13*n+9+(4*n^2-13*n+9)*x+(n^2-3*n+2)*x^2)*P(n,x) = -(16*n^3-88*n^2+141*n-54+(24*n^3-122*n^2+178*n-54)*x+(12*n^3-57*n^2+76*n-18)*x^2+(2*n^3-9*n^2+11*n-2)*x^3)*P(n-1,x) + (24*n^2-60*n+27+(56*n^2-134*n+54)*x+(46*n^2-104*n+36)*x^2+(16*n^2-34*n+10)*x^3+(2*n^2-4*n+1)*x^4)*d/dx 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[(-(16*n^3 - 88*n^2 + 141*n - 54 + (24*n^3 - 122*n^2 + 178*n - 54)*x + (12*n^3 - 57*n^2 + 76*n - 18)*x^2 + (2*n^3 - 9*n^2 + 11*n - 2)*x^3)*P[n - 1, x] + (24*n^2 - 60*n + 27 + (56*n^2 - 134*n + 54)*x + (46*n^2 - 104*n + 36)*x^2 + (16*n^2 - 34*n + 10)*x^3 + (2*n^2 - 4*n + 1)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 13*n + 9 + (4*n^2 - 13*n + 9)*x + (n^2 - 3*n + 2)*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] = (-(16*n^3 - 88*n^2 + 141*n - 54 + (24*n^3 - 122*n^2 + 178*n - 54)*x + (12*n^3 - 57*n^2 + 76*n - 18)*x^2 + (2*n^3 - 9*n^2 + 11*n - 2)*x^3)*v[i - 1] + (24*n^2 - 60*n + 27 + (56*n^2 - 134*n + 54)*x + (46*n^2 - 104*n + 36)*x^2 + (16*n^2 - 34*n + 10)*x^3 + (2*n^2 - 4*n + 1)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 13*n + 9 + (4*n^2 - 13*n + 9)*x + (n^2 - 3*n + 2)*x^2)); 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-7196be931c2ed996","steps":["reverse","D"],"edges":["e-67f427bf8180eb6f","e-b2bfd230baf53a23"],"text":"A152842 \u2192 reverse \u2192 D"},"headline":"A152842 \u2192 reverse \u2192 D","chain":null}