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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":"593c7dc25633e9490edc9e753816c3c519503645749be36e8c00bb19231c78f5","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-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 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:      4;\n  2:     36,       32;\n  3:    168,      512,      192;\n  4:    808,     3968,     4416,      1024;\n  5:   3420,    23680,    48960,     30720,      5120;\n  6:  14268,   126880,   381120,    445440,    189440,     24576;\n  7:  57232,   623104,  2478720,   4382720,   3399680,   1081344,    114688;\n  8: 225936,  2909184, 14408064,  34375680,  40796160,  23150592,   5849088,   524288;\n  9: 876276, 13030656, 77357952, 233607168, 380037120, 329121792, 145539072, 30408704, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3+(6*n^2-6*n)*x-(4*n-4)*x^2)*P(n,x) = -(18*n^2-75*n+51+(66*n^2-318*n+240)*x-(116*n-172)*x^2+32*x^3)*P(n-1,x) + (9*n-9+(72*n-78)*x+(150*n-198)*x^2+(24*n-124)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 75*n + 51 + (66*n^2 - 318*n + 240)*x - (116*n - 172)*x^2 + 32*x^3)*P[n - 1, x] + (9*n - 9 + (72*n - 78)*x + (150*n - 198)*x^2 + (24*n - 124)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n - 3 + (6*n^2 - 6*n)*x - (4*n - 4)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 - 75*n + 51 + (66*n^2 - 318*n + 240)*x - (116*n - 172)*x^2 + 32*x^3)*v[i - 1] + (9*n - 9 + (72*n - 78)*x + (150*n - 198)*x^2 + (24*n - 124)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(3*n - 3 + (6*n^2 - 6*n)*x - (4*n - 4)*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 A013611","D"],"edges":["e-09b3cf169dcfe004","e-b0a423be0cb7fdbd"],"text":"A081580 \u2192 basis A013611 \u2192 D"},"headline":"A081580 \u2192 basis A013611 \u2192 D","chain":null}