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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":"1905ed0192530380a3a9b978dbe90ccbdb7a7d01fe9fc97edacaa62bc5693e85","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-57c5a2449091b6f8","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (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:   0;\n  1:   0;\n  2:   1;\n  3:   2;\n  4:   7,   2;\n  5:  16,   6;\n  6:  38,  24,   3;\n  7:  82,  68,  12;\n  8: 173, 186,  54,  4;\n  9: 352, 462, 180, 20;\n","recurrence_sentence":"The row polynomials satisfy (n^3-4*n^2+n+6+(4*n^2-17*n+18)*x+(4*n-8)*x^2)*P(n,x) = (3*n^3-12*n^2+8*n+3+(2*n^3+2*n^2-35*n+39)*x+(8*n^2-22*n+2)*x^2+(8*n-24)*x^3)*P(n-1,x) - (5*n^2+(9*n^2+20*n-5)*x+(4*n^2+36*n+11)*x^2+(16*n+32)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 12*n^2 + 8*n + 3 + (2*n^3 + 2*n^2 - 35*n + 39)*x + (8*n^2 - 22*n + 2)*x^2 + (8*n - 24)*x^3)*P[n - 1, x] - (5*n^2 + (9*n^2 + 20*n - 5)*x + (4*n^2 + 36*n + 11)*x^2 + (16*n + 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + n + 6 + (4*n^2 - 17*n + 18)*x + (4*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^3 - 12*n^2 + 8*n + 3 + (2*n^3 + 2*n^2 - 35*n + 39)*x + (8*n^2 - 22*n + 2)*x^2 + (8*n - 24)*x^3)*v[i - 1] - (5*n^2 + (9*n^2 + 20*n - 5)*x + (4*n^2 + 36*n + 11)*x^2 + (16*n + 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + n + 6 + (4*n^2 - 17*n + 18)*x + (4*n - 8)*x^2)); 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-7d332fcb7bf9a0f7","steps":["D","D+1"],"edges":["e-44388154107108dc","e-2f04497a92816613"],"text":"A128100 \u2192 D \u2192 D+1"},"headline":"A128100 \u2192 D \u2192 D+1","chain":null}