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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":"f413018db596e8cc1a2a052f43d5deb8970bc8f2219e7fa3ea542a294a0a3e5f","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-4f8c7533ab8b37bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,5,1) array.","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:     42,     32,       3;\n  4:    202,    248,      69,      4;\n  5:    855,   1480,     765,    120,      5;\n  6:   3567,   7930,    5955,   1740,    185,     6;\n  7:  14308,  38944,   38730,  17120,   3320,   264,    7;\n  8:  56484, 181824,  225126, 134280,  39840,  5652,  357,   8;\n  9: 219069, 814416, 1208718, 912528, 371130, 80352, 8883, 464, 9;\n","recurrence_sentence":"The row polynomials satisfy (12*n-12+(6*n^2-6*n)*x-(n-1)*x^2)*P(n,x) = -(72*n^2-300*n+204+(66*n^2-318*n+240)*x-(29*n-43)*x^2+2*x^3)*P(n-1,x) + (144*n-144+(288*n-312)*x+(150*n-198)*x^2+(6*n-31)*x^3-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[(-(72*n^2 - 300*n + 204 + (66*n^2 - 318*n + 240)*x - (29*n - 43)*x^2 + 2*x^3)*P[n - 1, x] + (144*n - 144 + (288*n - 312)*x + (150*n - 198)*x^2 + (6*n - 31)*x^3 - x^4)*D[P[n - 1, x], x])/(12*n - 12 + (6*n^2 - 6*n)*x - (n - 1)*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] = (-(72*n^2 - 300*n + 204 + (66*n^2 - 318*n + 240)*x - (29*n - 43)*x^2 + 2*x^3)*v[i - 1] + (144*n - 144 + (288*n - 312)*x + (150*n - 198)*x^2 + (6*n - 31)*x^3 - x^4)*deriv(v[i - 1], x))/(12*n - 12 + (6*n^2 - 6*n)*x - (n - 1)*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-6ad01dc6e705cdb6","steps":["basis A007318","D"],"edges":["e-1ae1e4b2be227437","e-70896b100eb52b82"],"text":"A081580 \u2192 basis A007318 \u2192 D"},"headline":"A081580 \u2192 basis A007318 \u2192 D","chain":null}