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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":"d3165482a0e47692fceedea1badb220b2bd37bdba822f1b2f544b2090fa04dad","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f268708db386b729","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    0;\n   2:    1;\n   3:    5,     2;\n   4:   16,    18,     3;\n   5:   44,    82,    42,     4;\n   6:  112,   292,   255,    80,     5;\n   7:  272,   912,  1131,   620,   135,     6;\n   8:  640,  2624,  4224,  3332,  1295,   210,    7;\n   9: 1472,  7136, 14136, 14596,  8260,  2436,  308,   8;\n  10: 3328, 18624, 43824, 56008, 41805, 18144, 4242, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-16*n^2+27*n-14+(4*n^3-22*n^2+38*n-20)*x)*P(n,x) = (6*n^3-26*n^2+36*n-16+(17*n^3-82*n^2+121*n-56)*x+(12*n^3-66*n^2+116*n-70)*x^2)*P(n-1,x) - ((6*n^2-14*n+8)*x+(14*n^2-38*n+28)*x^2+(8*n^2-24*n+20)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^3 - 26*n^2 + 36*n - 16 + (17*n^3 - 82*n^2 + 121*n - 56)*x + (12*n^3 - 66*n^2 + 116*n - 70)*x^2)*P[n - 1, x] - ((6*n^2 - 14*n + 8)*x + (14*n^2 - 38*n + 28)*x^2 + (8*n^2 - 24*n + 20)*x^3)*D[P[n - 1, x], x])/(3*n^3 - 16*n^2 + 27*n - 14 + (4*n^3 - 22*n^2 + 38*n - 20)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((6*n^3 - 26*n^2 + 36*n - 16 + (17*n^3 - 82*n^2 + 121*n - 56)*x + (12*n^3 - 66*n^2 + 116*n - 70)*x^2)*v[i - 1] - ((6*n^2 - 14*n + 8)*x + (14*n^2 - 38*n + 28)*x^2 + (8*n^2 - 24*n + 20)*x^3)*deriv(v[i - 1], x))/(3*n^3 - 16*n^2 + 27*n - 14 + (4*n^3 - 22*n^2 + 38*n - 20)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2821c21537bbbcdd","steps":["reverse","D"],"edges":["e-23a5df9ffca078bb","e-b542c84abf101cb6"],"text":"A056242 \u2192 reverse \u2192 D"},"headline":"A056242 \u2192 reverse \u2192 D","chain":null}