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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":"d3200ecd3a37f63e9dc38c39acd45960d9aefc8ec7fe201985ffad1eb754d50f","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-0cecde7ff045a546","role":"sequence","name":"Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.","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:    9,     8;\n  3:   21,    48,     15;\n  4:   45,   200,    150,      24;\n  5:   93,   720,    975,     360,     35;\n  6:  189,  2408,   5250,    3360,    735,     48;\n  7:  381,  7728,  25515,   25200,   9310,   1344,    63;\n  8:  765, 24200, 116550,  166824,  92610,  22176,  2268,   80;\n  9: 1533, 74640, 511575, 1020600, 798945, 282240, 47250, 3600, 99;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = -(4+10*x+18*x^2-4*x^3)*P(n-3,x) + (9+x-17*x^2)*d/dx P(n-3,x) + (3*x+x^2-4*x^3)*d^2/dx^2 P(n-3,x) + (18+12*x-12*x^2)*P(n-2,x) + (6+16*x)*d/dx P(n-2,x) + (3*x+4*x^2)*d^2/dx^2 P(n-2,x) + (6+12*x)*P(n-1,x) - 3*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 9 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[2, x_] := 9 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4 + 10*x + 18*x^2 - 4*x^3)*P[n - 3, x] + (9 + x - 17*x^2)*D[P[n - 3, x], x] + (3*x + x^2 - 4*x^3)*D[P[n - 3, x], {x, 2}] + (18 + 12*x - 12*x^2)*P[n - 2, x] + (6 + 16*x)*D[P[n - 2, x], x] + (3*x + 4*x^2)*D[P[n - 2, x], {x, 2}] + (6 + 12*x)*P[n - 1, x] - 3*D[P[n - 1, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 9 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4 + 10*x + 18*x^2 - 4*x^3)*v[i - 3] + (9 + x - 17*x^2)*deriv(v[i - 3], x) + (3*x + x^2 - 4*x^3)*deriv(deriv(v[i - 3], x), x) + (18 + 12*x - 12*x^2)*v[i - 2] + (6 + 16*x)*deriv(v[i - 2], x) + (3*x + 4*x^2)*deriv(deriv(v[i - 2], x), x) + (6 + 12*x)*v[i - 1] - 3*deriv(v[i - 1], 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-3cb22164e2f83fe1","steps":["\u03b8+2 (shift +1)","D"],"edges":["e-4ce9900b2631a004","e-967e3499abfd84e8"],"text":"A008277 \u2192 \u03b8+2 (shift +1) \u2192 D"},"headline":"A008277 \u2192 \u03b8+2 (shift +1) \u2192 D","chain":null}