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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":"9499d0581337a845e88ef2f40f16020dc38b95da47a469b8684d53b8fb80ea84","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-2779422d3a9a18c1","role":"sequence","name":"shift_x_plus_one 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:      2;\n   3:     13,       8;\n   4:     65,      80,      24;\n   5:    292,     538,     324,      64;\n   6:   1236,    3030,    2742,    1088,     160;\n   7:   5036,   15404,   18603,   11104,    3280,     384;\n   8:  19980,   73228,  110595,   88192,   39200,    9216,    896;\n   9:  77728,  331952,  601752,  600676,  356840,  126240,  24640,  2048;\n  10: 297856, 1452304, 3071832, 3683532, 2740410, 1295904, 380576, 63488, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n^3+2*n^2-13*n+10+(4*n^2-12*n+8)*x)*P(n,x) = (n^3+10*n^2-5*n+2+(10*n^2+14*n-16)*x+(16*n-16)*x^2)*P(n-1,x) + (4*n^2+12*n-8+(6*n^2+34*n-28)*x+(2*n^2+30*n-28)*x^2+(8*n-8)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 10*n^2 - 5*n + 2 + (10*n^2 + 14*n - 16)*x + (16*n - 16)*x^2)*P[n - 1, x] + (4*n^2 + 12*n - 8 + (6*n^2 + 34*n - 28)*x + (2*n^2 + 30*n - 28)*x^2 + (8*n - 8)*x^3)*D[P[n - 1, x], x])/(n^3 + 2*n^2 - 13*n + 10 + (4*n^2 - 12*n + 8)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((n^3 + 10*n^2 - 5*n + 2 + (10*n^2 + 14*n - 16)*x + (16*n - 16)*x^2)*v[i - 1] + (4*n^2 + 12*n - 8 + (6*n^2 + 34*n - 28)*x + (2*n^2 + 30*n - 28)*x^2 + (8*n - 8)*x^3)*deriv(v[i - 1], x))/(n^3 + 2*n^2 - 13*n + 10 + (4*n^2 - 12*n + 8)*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":["x \u21a6 x+1","D"],"edges":["e-991bb8f582af0f63","e-4b2421445cb9df9f"],"text":"A056242 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A056242 \u2192 x \u21a6 x+1 \u2192 D","chain":null}