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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":"2935499e812192e4abe79b27cdde7d9c4406e80b1a0b7ad842a4fc6d866d9039","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-9813839be9bee901","role":"sequence","name":"basis_map_A013610 of Pascal-(1,3,1) 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:     21,      18;\n  3:     90,     216,       81;\n  4:    366,    1404,     1377,      324;\n  5:   1365,    7272,    12231,     7128,     1215;\n  6:   4923,   33534,    80271,    80676,    32805,     4374;\n  7:  17220,  142992,   446310,   650592,   451980,   139968,    15309;\n  8:  59052,  578520,  2229606,  4291704,  4349700,  2274480,   566433,   52488;\n  9: 199287, 2250288, 10331874, 24809328, 33468390, 25509168, 10609137, 2204496, 177147;\n","recurrence_sentence":"The row polynomials satisfy (24*n-24+(36*n^2-36*n)*x-(27*n-27)*x^2)*P(n,x) = -(96*n^2-408*n+264+(252*n^2-1332*n+1008)*x-(513*n-783)*x^2+162*x^3)*P(n-1,x) + (64*n-64+(384*n-432)*x+(612*n-900)*x^2+(108*n-567)*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[(-(96*n^2 - 408*n + 264 + (252*n^2 - 1332*n + 1008)*x - (513*n - 783)*x^2 + 162*x^3)*P[n - 1, x] + (64*n - 64 + (384*n - 432)*x + (612*n - 900)*x^2 + (108*n - 567)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(24*n - 24 + (36*n^2 - 36*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] = (-(96*n^2 - 408*n + 264 + (252*n^2 - 1332*n + 1008)*x - (513*n - 783)*x^2 + 162*x^3)*v[i - 1] + (64*n - 64 + (384*n - 432)*x + (612*n - 900)*x^2 + (108*n - 567)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(24*n - 24 + (36*n^2 - 36*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-d8f9ad9ce0e039d0","steps":["basis A013610","D"],"edges":["e-a15ab36c8834192d","e-c686ababc2f60f15"],"text":"A081578 \u2192 basis A013610 \u2192 D"},"headline":"A081578 \u2192 basis A013610 \u2192 D","chain":null}