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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":"a6e68cd77d51fbf0eb1d999f7f1ff926c69458cfe14c77c285049d994f42f7f3","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:      1;\n   2:      5,       2;\n   3:     23,      21,       4;\n   4:     99,     145,      64,       8;\n   5:    408,     830,     593,     172,      16;\n   6:   1632,    4266,    4257,    2002,     432,      32;\n   7:   6388,   20440,   26305,   17305,    6056,    1040,     64;\n   8:  24596,   93208,  147209,  125057,   61248,   17056,   2432,    128;\n   9:  93488,  409680,  767728,  801260,  507009,  197608,  45680,   5568,   256;\n  10: 351664, 1750160, 3797984, 4707476, 3661293, 1843986, 596560, 117856, 12544, 512;\n","recurrence_sentence":"The row polynomials satisfy (n^3-6*n^2+15*n-10-(8*n^2-18*n+10)*x-(4*n^2-4)*x^2-(4*n-4)*x^3)*P(n,x) = (n^3+2*n^2+7*n-10-(2*n^2-4*n+22)*x-(4*n^2+16*n+24)*x^2-(12*n+20)*x^3-8*x^4)*P(n-1,x) + (4*n^2-20*n-24+(6*n^2-62*n-92)*x+(2*n^2-74*n-144)*x^2-(40*n+116)*x^3-(8*n+48)*x^4-8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 2*n^2 + 7*n - 10 - (2*n^2 - 4*n + 22)*x - (4*n^2 + 16*n + 24)*x^2 - (12*n + 20)*x^3 - 8*x^4)*P[n - 1, x] + (4*n^2 - 20*n - 24 + (6*n^2 - 62*n - 92)*x + (2*n^2 - 74*n - 144)*x^2 - (40*n + 116)*x^3 - (8*n + 48)*x^4 - 8*x^5)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 15*n - 10 - (8*n^2 - 18*n + 10)*x - (4*n^2 - 4)*x^2 - (4*n - 4)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^3 + 2*n^2 + 7*n - 10 - (2*n^2 - 4*n + 22)*x - (4*n^2 + 16*n + 24)*x^2 - (12*n + 20)*x^3 - 8*x^4)*v[i - 1] + (4*n^2 - 20*n - 24 + (6*n^2 - 62*n - 92)*x + (2*n^2 - 74*n - 144)*x^2 - (40*n + 116)*x^3 - (8*n + 48)*x^4 - 8*x^5)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 15*n - 10 - (8*n^2 - 18*n + 10)*x - (4*n^2 - 4)*x^2 - (4*n - 4)*x^3)); 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+1"],"edges":["e-991bb8f582af0f63","e-2f90d5c515873f5a"],"text":"A056242 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A056242 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}