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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":"17343e3944bc15bcd17adfc02b0e9d02c4bca5b611f3f4d97d041535df9a5722","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-0990b5be9f344750","role":"sequence","name":"Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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,    2;\n  2:  6,   13,     4;\n  3: 10,   47,    43,      8;\n  4: 15,  124,   250,    129,     16;\n  5: 21,  270,   980,   1129,    371,      32;\n  6: 28,  517,  2975,   6314,   4676,    1049,     64;\n  7: 36,  903,  7574,  25984,  36246,   18423,   2955,    128;\n  8: 45, 1472, 16968,  86142, 198051,  193584,  70470,   8353,   256;\n  9: 55, 2274, 34500, 243558, 841407, 1381890, 985720, 265013, 23779, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+2*x)*P(n,x) = -(n+1+2*n^2*x+(n^3+4*n-3)*x^2+(3*n^2-n+2)*x^3+2*n*x^4)*P(n-2,x) + ((n+1)*x^2+(n^2+n+2)*x^3+(3*n+1)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n+1+(2*n^2+4)*x+(5*n-1)*x^2+2*x^3)*P(n-1,x) - (n*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + 2*n^2*x + (n^3 + 4*n - 3)*x^2 + (3*n^2 - n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + n + 2)*x^3 + (3*n + 1)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n + 1 + (2*n^2 + 4)*x + (5*n - 1)*x^2 + 2*x^3)*P[n - 1, x] - (n*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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