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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":"69d1065ad2a8730715047a11c399fda4b84c5a4fdee3ae84a2ba66e6a8dbd185","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-3951e6b9a140d595","role":"sequence","name":"Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).","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:         3;\n  2:        15,         3;\n  3:       105,        45;\n  4:       900,       630,        45;\n  5:      8820,      9450,      1575;\n  6:     95760,    151200,     42525,      1575;\n  7:   1134000,   2540160,   1091475,     99225;\n  8:  14515200,  44452800,  27783000,   4365900,   99225;\n  9: 199584000, 808315200, 707275800, 170270100, 9823275;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+10+(4*n-14)*x+4*x^2)*P(n,x) = (n^3+n^2+6*n+16+(n^3+n-32)*x+(4*n^2-10*n+18)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((2*n^2-2*n+8)*x-(2*n^2-10*n+20)*x^2-(8*n-20)*x^3-8*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^3 + n^2 + 6*n + 16 + (n^3 + n - 32)*x + (4*n^2 - 10*n + 18)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 8)*x - (2*n^2 - 10*n + 20)*x^2 - (8*n - 20)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^2 - 3*n + 10 + (4*n - 14)*x + 4*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^3 + n^2 + 6*n + 16 + (n^3 + n - 32)*x + (4*n^2 - 10*n + 18)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((2*n^2 - 2*n + 8)*x - (2*n^2 - 10*n + 20)*x^2 - (8*n - 20)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^2 - 3*n + 10 + (4*n - 14)*x + 4*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-3951e6b9a140d595","steps":["D+1"],"edges":["e-278e14ac7185778e"],"text":"A306364 \u2192 D+1"},"headline":"A306364 \u2192 D+1","chain":null}