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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"0f906d0b852278c42132566ff56df89d9a44b87082df661f0d17d87e42f94e11","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-be7df2e3fd154f3c","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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,            4;\n  2:          59,          108,           32;\n  3:         593,         1992,         1904,           384;\n  4:        7017,        35224,        61040,         40064,          6144;\n  5:       96087,       649244,      1667440,       1971264,        991744,         122880;\n  6:     1498611,     12802868,     44021200,      76730304,      69259264,       28348416,       2949120;\n  7:    26262945,    272181456,   1178949792,    2741028864,    3635079424,     2662789120,     920567808,      82575360;\n  8:   511334865,   6243048432,  32719081632,   95576972288,  168624232704,   181031473152,  111820193792,   33504362496,    2642411520;\n  9: 10955901735, 154193491188, 950203408416, 3351881353856, 7415316188928, 10570027889664, 9551245967360, 5107252396032, 1351489683456, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(16*n^3-24*n^2+8*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 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-7fd16a32c438d522","steps":["basis A013611","D+1"],"edges":["e-5d9e378754634336","e-3ce287ac739dc74d"],"text":"A059364 \u2192 basis A013611 \u2192 D+1"},"headline":"A059364 \u2192 basis A013611 \u2192 D+1","chain":null}