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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":"93c88ba2d4c899922387d6ed25aaf0d4ee652ebd6b3ff1fa6d8658bb3c3b1eaf","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-fefc87d068bda822","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","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:              1,               1;\n  2:              1,               3,               1;\n  3:              8,              26,              12,              1;\n  4:            216,             718,             359,             40,            1;\n  5:          13824,           46384,           23945,           2955,          105,           1;\n  6:        1728000,         5825648,         3055789,         395875,        16180,         231,         1;\n  7:      373248000,      1261795968,       667924896,       88900444,      3905810,       66301,       448,      1;\n  8:   128024064000,    433542513024,    230804309280,    31235327492,   1432180809,    26710708,    220406,    792,    1;\n  9: 65548320768000, 222229814796288, 118758108088896, 16249307013440, 765816976509, 15133484505, 139773498, 626694, 1305, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(2*n^3-12*n^2+24*n-16)*x+x^2)*P(n-2,x) + (2*n^3-9*n^2+15*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*P[n - 2, x] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*v[i - 2] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*v[i - 1]); 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-fefc87d068bda822","steps":["D+1"],"edges":["e-fad3c72cd2bd3d8b"],"text":"A269947 \u2192 D+1"},"headline":"A269947 \u2192 D+1","chain":null}