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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":"3bc69affd6ba28b01424b56494578fe4aef406d909f24eb317377273920fd4be","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-2ee89feef605b3fb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,9,1) 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:       1;\n  1:       3,       1;\n  2:      26,      15,       1;\n  3:     110,     114,      27,       1;\n  4:     615,     890,     345,      39,       1;\n  5:    2821,    5415,    3305,     719,      51,      1;\n  6:   13828,   32901,   27000,    9049,    1236,     63,     1;\n  7:   63756,  182260,  191946,   91374,   19816,   1896,    75,    1;\n  8:  297245,  991764, 1272982,  788210,  244506,  37300,  2699,   87,  1;\n  9: 1355255, 5179005, 7879590, 6074778, 2511702, 552330, 63195, 3645, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (20*n^2+20*n+(10*n^3+30*n^2+20*n)*x+(39*n^2+19*n)*x^2+(n^2+38*n)*x^3+n*x^4)*P(n,x) = -(200*n^3-20*n^2-660*n+360+(190*n^3+710*n^2-1080*n-40)*x+(731*n^2+253*n-1218)*x^2+(19*n^2+683*n-877)*x^3+(18*n-60)*x^4-x^5)*P(n-1,x) + (400*n^2+1200*n+800+(800*n^2+3960*n+3960)*x+(410*n^2+4390*n+7140)*x^2+(10*n^2+1709*n+5679)*x^3+(80*n+1779)*x^4+(n+81)*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[(-(200*n^3 - 20*n^2 - 660*n + 360 + (190*n^3 + 710*n^2 - 1080*n - 40)*x + (731*n^2 + 253*n - 1218)*x^2 + (19*n^2 + 683*n - 877)*x^3 + (18*n - 60)*x^4 - x^5)*P[n - 1, x] + (400*n^2 + 1200*n + 800 + (800*n^2 + 3960*n + 3960)*x + (410*n^2 + 4390*n + 7140)*x^2 + (10*n^2 + 1709*n + 5679)*x^3 + (80*n + 1779)*x^4 + (n + 81)*x^5 + x^6)*D[P[n - 1, x], x])/(20*n^2 + 20*n + (10*n^3 + 30*n^2 + 20*n)*x + (39*n^2 + 19*n)*x^2 + (n^2 + 38*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] = (-(200*n^3 - 20*n^2 - 660*n + 360 + (190*n^3 + 710*n^2 - 1080*n - 40)*x + (731*n^2 + 253*n - 1218)*x^2 + (19*n^2 + 683*n - 877)*x^3 + (18*n - 60)*x^4 - x^5)*v[i - 1] + (400*n^2 + 1200*n + 800 + (800*n^2 + 3960*n + 3960)*x + (410*n^2 + 4390*n + 7140)*x^2 + (10*n^2 + 1709*n + 5679)*x^3 + (80*n + 1779)*x^4 + (n + 81)*x^5 + x^6)*deriv(v[i - 1], x))/(20*n^2 + 20*n + (10*n^3 + 30*n^2 + 20*n)*x + (39*n^2 + 19*n)*x^2 + (n^2 + 38*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-7ac21ea1b3b7e282","steps":["basis A007318","D+1"],"edges":["e-a3fef8dca23d39bb","e-0f27c52fc111faaf"],"text":"A143685 \u2192 basis A007318 \u2192 D+1"},"headline":"A143685 \u2192 basis A007318 \u2192 D+1","chain":null}