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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":"55faca5120ad94f4bcad5da8ddfd62fac905b04a4238120946eda993c2383355","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-d07b286c67e2d8fc","role":"sequence","name":"basis_map_A013610 of Pascal-(1,4,1) 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:      3;\n  2:     24,      18;\n  3:    108,     252,       81;\n  4:    480,    1800,     1620,      324;\n  5:   1920,   10080,    16200,     8424,     1215;\n  6:   7488,   50400,   116640,   108864,    38880,     4374;\n  7:  28224,  232128,   706320,   977184,   617220,   166212,    15309;\n  8: 104448, 1014912,  3830976,  7081344,  6687360,  3131784,   673596,   52488;\n  9: 380160, 4262400, 19232640, 44696448, 57056400, 39890880, 14696640, 2624400, 177147;\n","recurrence_sentence":"The row polynomials satisfy (30*n-30+(45*n^2-45*n)*x-(27*n-27)*x^2)*P(n,x) = -(150*n^2-630*n+420+(405*n^2-2025*n+1530)*x-(648*n-972)*x^2+162*x^3)*P(n-1,x) + (100*n-100+(600*n-660)*x+(945*n-1305)*x^2+(135*n-702)*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(150*n^2 - 630*n + 420 + (405*n^2 - 2025*n + 1530)*x - (648*n - 972)*x^2 + 162*x^3)*P[n - 1, x] + (100*n - 100 + (600*n - 660)*x + (945*n - 1305)*x^2 + (135*n - 702)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(30*n - 30 + (45*n^2 - 45*n)*x - (27*n - 27)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(150*n^2 - 630*n + 420 + (405*n^2 - 2025*n + 1530)*x - (648*n - 972)*x^2 + 162*x^3)*v[i - 1] + (100*n - 100 + (600*n - 660)*x + (945*n - 1305)*x^2 + (135*n - 702)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(30*n - 30 + (45*n^2 - 45*n)*x - (27*n - 27)*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-ed9434bfd889c409","steps":["basis A013610","D"],"edges":["e-14634eb9644b375c","e-875e1ffd425511d8"],"text":"A081579 \u2192 basis A013610 \u2192 D"},"headline":"A081579 \u2192 basis A013610 \u2192 D","chain":null}