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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":"1ebe030f5a666540a7cd7e74e57d412ec7bcfea2506cbaf3613a737ebc4b4842","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-3bdf31527eaa2fb3","role":"sequence","name":"basis_map_A007318 of Pascal-(1,2,1) array read by antidiagonals.","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:     3,      1;\n  2:    12,      8,      1;\n  3:    40,     44,     13,      1;\n  4:   132,    204,    100,     18,      1;\n  5:   420,    844,    596,    180,     23,     1;\n  6:  1312,   3256,   3032,   1328,    284,    28,    1;\n  7:  4032,  11936,  13880,   8088,   2512,   412,   33,   1;\n  8: 12240,  42144,  58944,  43200,  17916,  4260,  564,  38,  1;\n  9: 36784, 144528, 236640, 209856, 109932, 34932, 6684, 740, 43, 1;\n","recurrence_sentence":"The row polynomials satisfy (6*n^2+6*n+(3*n^3+9*n^2+6*n)*x+(11*n^2+5*n)*x^2+(n^2+10*n)*x^3+n*x^4)*P(n,x) = -(18*n^3-6*n^2-72*n+24+(15*n^3+45*n^2-114*n-12)*x+(52*n^2-6*n-112)*x^2+(5*n^2+39*n-93)*x^3+(4*n-18)*x^4-x^5)*P(n-1,x) + (36*n^2+108*n+72+(72*n^2+348*n+348)*x+(39*n^2+393*n+630)*x^2+(3*n^2+176*n+527)*x^3+(24*n+197)*x^4+(n+25)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(18*n^3 - 6*n^2 - 72*n + 24 + (15*n^3 + 45*n^2 - 114*n - 12)*x + (52*n^2 - 6*n - 112)*x^2 + (5*n^2 + 39*n - 93)*x^3 + (4*n - 18)*x^4 - x^5)*P[n - 1, x] + (36*n^2 + 108*n + 72 + (72*n^2 + 348*n + 348)*x + (39*n^2 + 393*n + 630)*x^2 + (3*n^2 + 176*n + 527)*x^3 + (24*n + 197)*x^4 + (n + 25)*x^5 + x^6)*D[P[n - 1, x], x])/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (11*n^2 + 5*n)*x^2 + (n^2 + 10*n)*x^3 + n*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(18*n^3 - 6*n^2 - 72*n + 24 + (15*n^3 + 45*n^2 - 114*n - 12)*x + (52*n^2 - 6*n - 112)*x^2 + (5*n^2 + 39*n - 93)*x^3 + (4*n - 18)*x^4 - x^5)*v[i - 1] + (36*n^2 + 108*n + 72 + (72*n^2 + 348*n + 348)*x + (39*n^2 + 393*n + 630)*x^2 + (3*n^2 + 176*n + 527)*x^3 + (24*n + 197)*x^4 + (n + 25)*x^5 + x^6)*deriv(v[i - 1], x))/(6*n^2 + 6*n + (3*n^3 + 9*n^2 + 6*n)*x + (11*n^2 + 5*n)*x^2 + (n^2 + 10*n)*x^3 + n*x^4)); 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-f25cd955a876cde3","steps":["x \u21a6 x+1","D+1"],"edges":["e-6f8e74d475c3b6cc","e-ffea75e78cbb9f2d"],"text":"A081577 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A081577 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}