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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":"47321d83b633730cd76a81bee50a7dbfa7231be168a0e2fe356ca552d1115ab7","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-081b3f700db2d097","role":"sequence","name":"veronese_section_2_0 of Cube of lower triangular normalized binomial matrix.","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;\n  2:    10,      1;\n  3:    36,      9;\n  4:   135,     56,     1;\n  5:   513,    300,    15;\n  6:  1944,   1485,   138,    1;\n  7:  7290,   6993,  1008,   21;\n  8: 26973,  31752,  6426,  256,  1;\n  9: 98415, 139968, 37422, 2376, 27;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-12*n^2+69*n-60+(12*n^2-150*n+138)*x+(12*n-12)*x^2)*P(n,x) = (9*n^3-18*n^2+171*n-162+(n^3+32*n^2-465*n+432)*x+(4*n^2-10*n+6)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((18*n^2-18*n-324)*x-(2*n^2-74*n+648)*x^2-(8*n-148)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 18*n^2 + 171*n - 162 + (n^3 + 32*n^2 - 465*n + 432)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((18*n^2 - 18*n - 324)*x - (2*n^2 - 74*n + 648)*x^2 - (8*n - 148)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 69*n - 60 + (12*n^2 - 150*n + 138)*x + (12*n - 12)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 18*n^2 + 171*n - 162 + (n^3 + 32*n^2 - 465*n + 432)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((18*n^2 - 18*n - 324)*x - (2*n^2 - 74*n + 648)*x^2 - (8*n - 148)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 69*n - 60 + (12*n^2 - 150*n + 138)*x + (12*n - 12)*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-8d887bfc40d036b9","steps":["Veronese 2","D+1"],"edges":["e-5d982f8f22366510","e-20080495ea622ee9"],"text":"A027465 \u2192 Veronese 2 \u2192 D+1"},"headline":"A027465 \u2192 Veronese 2 \u2192 D+1","chain":null}