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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":"8c3b4d3ab516df00869c0d4145a193fa2002fe1f34c9e4da88f6ef1018529707","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-7e652d8a57b6dcff","role":"sequence","name":"Triangle of 4-Eulerian numbers.","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:    5,      4;\n  2:   14,     45,       16;\n  3:   33,    258,      305,        64;\n  4:   72,   1133,     2994,      1845,       256;\n  5:  151,   4330,    21500,     28010,     10505,      1024;\n  6:  310,  15249,   129000,    303490,    231090,     57645,      4096;\n  7:  629,  51010,   690591,   2670980,   3576715,   1756986,    308705,    16384;\n  8: 1268, 164973,  3423610,  20457419,  44052720,  37335907,  12623522,  1625445,   65536;\n  9: 2547, 521538, 16082114, 142092790, 462940716, 621616478, 357424102, 87066666, 8454905, 262144;\n","recurrence_sentence":"The row polynomials satisfy (n+1+2*x)*P(n,x) = -(2*n+4+(2*n^2+5*n+6)*x+(n^3+4*n^2+9*n+4)*x^2+(3*n^2+8*n+6)*x^3+(2*n+4)*x^4)*P(n-2,x) - ((n+2)*x+(n^2+2*n+2)*x^2-n^2*x^3-(3*n+2)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n+4+(2*n^2+5*n+9)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) + ((n+1)*x-(n-1)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (2*n^2 + 5*n + 6)*x + (n^3 + 4*n^2 + 9*n + 4)*x^2 + (3*n^2 + 8*n + 6)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((n + 2)*x + (n^2 + 2*n + 2)*x^2 - n^2*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 4 + (2*n^2 + 5*n + 9)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (2*n^2 + 5*n + 6)*x + (n^3 + 4*n^2 + 9*n + 4)*x^2 + (3*n^2 + 8*n + 6)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((n + 2)*x + (n^2 + 2*n + 2)*x^2 - n^2*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 4 + (2*n^2 + 5*n + 9)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*x)); 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-7e652d8a57b6dcff","steps":["D+1"],"edges":["e-f5db20c0d0fdad70"],"text":"A144698 \u2192 D+1"},"headline":"A144698 \u2192 D+1","chain":null}