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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":"7c1b3770962734f573b6c36e047e4a559d917ab907f9f8a6a3a260204216379e","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-f2968a7a3d3a43cd","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (2+3x)^n.","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:     2;\n  2:    13,      9;\n  3:    62,     54;\n  4:   232,    378,      81;\n  5:   752,   2340,     810;\n  6:  2224,  11880,    7047,    729;\n  7:  6176,  51408,   53298,  10206;\n  8: 16384, 197568,  335664, 107892,   6561;\n  9: 41984, 694656, 1796256, 962280, 118098;\n","recurrence_sentence":"The row polynomials satisfy (18*n^3-72*n^2+106*n-52+(72*n^2-284*n+212)*x+(72*n-72)*x^2)*P(n,x) = (36*n^3-72*n^2+68*n-32+(81*n^3-180*n^2-397*n+496)*x+(324*n^2-810*n+486)*x^2+(324*n-324)*x^3)*P(n-1,x) + ((72*n^2-72*n-64)*x-(162*n^2-450*n+128)*x^2-(648*n-900)*x^3-648*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((36*n^3 - 72*n^2 + 68*n - 32 + (81*n^3 - 180*n^2 - 397*n + 496)*x + (324*n^2 - 810*n + 486)*x^2 + (324*n - 324)*x^3)*P[n - 1, x] + ((72*n^2 - 72*n - 64)*x - (162*n^2 - 450*n + 128)*x^2 - (648*n - 900)*x^3 - 648*x^4)*D[P[n - 1, x], x])/(18*n^3 - 72*n^2 + 106*n - 52 + (72*n^2 - 284*n + 212)*x + (72*n - 72)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((36*n^3 - 72*n^2 + 68*n - 32 + (81*n^3 - 180*n^2 - 397*n + 496)*x + (324*n^2 - 810*n + 486)*x^2 + (324*n - 324)*x^3)*v[i - 1] + ((72*n^2 - 72*n - 64)*x - (162*n^2 - 450*n + 128)*x^2 - (648*n - 900)*x^3 - 648*x^4)*deriv(v[i - 1], x))/(18*n^3 - 72*n^2 + 106*n - 52 + (72*n^2 - 284*n + 212)*x + (72*n - 72)*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-5733d79f45a14084","steps":["Veronese 2","D+1"],"edges":["e-bdf59334244a08a5","e-acf6de0a15c6487e"],"text":"A013620 \u2192 Veronese 2 \u2192 D+1"},"headline":"A013620 \u2192 Veronese 2 \u2192 D+1","chain":null}