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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":"785077f7f2df0300e9e84328c7fe71efe0948b3296d315fbe546631feb0639e7","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-e88a150cc5b6bb0f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=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:   1;\n  2:   2,     1;\n  3:   5,     4;\n  4:  12,    17,     3;\n  5:  27,    76,    25;\n  6:  58,   317,   175,    15;\n  7: 121,  1212,  1176,   210;\n  8: 248,  4321,  7287,  2170,  105;\n  9: 503, 14644, 41175, 20188, 2205;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(n^3-7*n^2+16*n-12)*x^2*P(n-4,x) - ((2*n^2-8*n+8)*x-(n^2-5*n+6)*x^2)*P(n-3,x) - (n^2-4*n+4)*x^2*d/dx P(n-3,x) - (2*n-4-(2*n^2-7*n+4)*x)*P(n-2,x) - ((n-2)*x-(n-3)*x^2)*d/dx P(n-2,x) + (3*n-8-(n-3)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 5 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 16*n - 12)*x^2*P[n - 4, x] - ((2*n^2 - 8*n + 8)*x - (n^2 - 5*n + 6)*x^2)*P[n - 3, x] - (n^2 - 4*n + 4)*x^2*D[P[n - 3, x], x] - (2*n - 4 - (2*n^2 - 7*n + 4)*x)*P[n - 2, x] - ((n - 2)*x - (n - 3)*x^2)*D[P[n - 2, x], x] + (3*n - 8 - (n - 3)*x)*P[n - 1, x] + (n - 3)*x*D[P[n - 1, x], x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 5 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - 7*n^2 + 16*n - 12)*x^2*v[i - 4] - ((2*n^2 - 8*n + 8)*x - (n^2 - 5*n + 6)*x^2)*v[i - 3] - (n^2 - 4*n + 4)*x^2*deriv(v[i - 3], x) - (2*n - 4 - (2*n^2 - 7*n + 4)*x)*v[i - 2] - ((n - 2)*x - (n - 3)*x^2)*deriv(v[i - 2], x) + (3*n - 8 - (n - 3)*x)*v[i - 1] + (n - 3)*x*deriv(v[i - 1], x))/(n - 3)); 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-e88a150cc5b6bb0f","steps":["D+1"],"edges":["e-3f61ea97669fe920"],"text":"A124324 \u2192 D+1"},"headline":"A124324 \u2192 D+1","chain":null}