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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":"6280f0a559a1134dc5b62f4e1e8221b6f59f358c33fff542653111a2d5daa7da","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9516b04042fe64c0","role":"sequence","name":"Coefficient triangle of certain polynomials N(4; m,x).","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:   6,    5;\n  2:  13,   42,    15;\n  3:  22,  147,   168,     35;\n  4:  33,  368,   840,    504,     70;\n  5:  46,  765,  2880,   3360,   1260,     126;\n  6:  61, 1410,  7875,  15000,  10710,    2772,    210;\n  7:  78, 2387, 18480,  51975,  60060,   29106,   5544,    330;\n  8:  97, 3792, 38808, 150920, 256410,  199584,  70224,  10296,   495;\n  9: 118, 5733, 74880, 384384, 900900, 1027026, 576576, 154440, 18018, 715;\n","recurrence_sentence":"The row polynomials satisfy (n^4+6*n^3+6*n^2-8*n+(2*n^3+8*n^2)*x+(n^2+4*n)*x^2)*P(n,x) = (n^4+8*n^3+17*n^2+4*n+(3*n^4+20*n^3+38*n^2+24*n)*x+(6*n^3+24*n^2+38*n+12)*x^2+(3*n^2+10*n+12)*x^3)*P(n-1,x) + ((2*n^3+11*n^2+12*n)*x-(2*n^3+7*n^2-8)*x^2-(4*n^2+10*n+4)*x^3-(2*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + 8*n^3 + 17*n^2 + 4*n + (3*n^4 + 20*n^3 + 38*n^2 + 24*n)*x + (6*n^3 + 24*n^2 + 38*n + 12)*x^2 + (3*n^2 + 10*n + 12)*x^3)*P[n - 1, x] + ((2*n^3 + 11*n^2 + 12*n)*x - (2*n^3 + 7*n^2 - 8)*x^2 - (4*n^2 + 10*n + 4)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 6*n^2 - 8*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 8*n^3 + 17*n^2 + 4*n + (3*n^4 + 20*n^3 + 38*n^2 + 24*n)*x + (6*n^3 + 24*n^2 + 38*n + 12)*x^2 + (3*n^2 + 10*n + 12)*x^3)*v[i - 1] + ((2*n^3 + 11*n^2 + 12*n)*x - (2*n^3 + 7*n^2 - 8)*x^2 - (4*n^2 + 10*n + 4)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 6*n^2 - 8*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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-9516b04042fe64c0","steps":["D+1"],"edges":["e-e09e0bff614306ec"],"text":"A062264 \u2192 D+1"},"headline":"A062264 \u2192 D+1","chain":null}