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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":"4a758422383f437bc5e9fb8cada12e6a252edead9faf94f69ebf3d00c9f783cc","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-5aa52e3acf29a627","role":"sequence","name":"Triangle read by rows, related to clusters of type D.","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:      7,       3;\n  2:     35,      32,       6;\n  3:    161,     225,      90,      10;\n  4:    714,    1344,     840,     200,      15;\n  5:   3102,    7350,    6300,    2380,     385,      21;\n  6:  13299,   38016,   41580,   21840,    5670,     672,     28;\n  7:  56485,  189189,  252252,  170940,   62370,   11970,   1092,    36;\n  8: 238238,  915200, 1441440, 1201200,  570570,  155232,  23100,  1680,   45;\n  9: 999362, 4332042, 7876440, 7807800, 4594590, 1639638, 348348, 41580, 2475, 55;\n","recurrence_sentence":"The row polynomials satisfy (n^4+4*n^3+3*n^2-2*n+(2*n^3+4*n^2)*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^4+18*n^3+16*n^2+4*n+(3*n^4+20*n^3+23*n^2+12*n+2)*x+(6*n^3+15*n^2+14*n+4)*x^2+(3*n^2+4*n+2)*x^3)*P(n-1,x) - ((2*n^3+9*n^2+8*n+2)*x+(2*n^3+13*n^2+16*n+6)*x^2+(4*n^2+10*n+6)*x^3+(2*n+2)*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[((4*n^4 + 18*n^3 + 16*n^2 + 4*n + (3*n^4 + 20*n^3 + 23*n^2 + 12*n + 2)*x + (6*n^3 + 15*n^2 + 14*n + 4)*x^2 + (3*n^2 + 4*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 + 9*n^2 + 8*n + 2)*x + (2*n^3 + 13*n^2 + 16*n + 6)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 4*n^3 + 3*n^2 - 2*n + (2*n^3 + 4*n^2)*x + (n^2 + 2*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] = ((4*n^4 + 18*n^3 + 16*n^2 + 4*n + (3*n^4 + 20*n^3 + 23*n^2 + 12*n + 2)*x + (6*n^3 + 15*n^2 + 14*n + 4)*x^2 + (3*n^2 + 4*n + 2)*x^3)*v[i - 1] - ((2*n^3 + 9*n^2 + 8*n + 2)*x + (2*n^3 + 13*n^2 + 16*n + 6)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 4*n^3 + 3*n^2 - 2*n + (2*n^3 + 4*n^2)*x + (n^2 + 2*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-5aa52e3acf29a627","steps":["D+1"],"edges":["e-3eb8cfb2829131e0"],"text":"A345013 \u2192 D+1"},"headline":"A345013 \u2192 D+1","chain":null}