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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":"33cf74060cf70fd4a563bfc25cfc627e76bf5447a32721796e5b905ba7c23850","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-0d89b6177f6a1dbb","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 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:    1;\n  1:    1;\n  2:    5,    2;\n  3:    9,    4;\n  4:   25,   22,    4;\n  5:   53,   56,   12;\n  6:  125,  178,   72,   8;\n  7:  273,  460,  232,  32;\n  8:  609, 1230,  828, 208, 16;\n  9: 1325, 3072, 2484, 800, 80;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+3*n^2-8*n+(8*n^2-3*n)*x+8*n*x^2)*P(n,x) = (10*n^3+9*n^2-34*n+12+(8*n^3+44*n^2-39*n-8)*x+(32*n^2+28*n-52)*x^2+(32*n-32)*x^3)*P(n-1,x) - (18*n^2+63*n+45+(34*n^2+191*n+202)*x+(16*n^2+192*n+333)*x^2+(64*n+240)*x^3+64*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((10*n^3 + 9*n^2 - 34*n + 12 + (8*n^3 + 44*n^2 - 39*n - 8)*x + (32*n^2 + 28*n - 52)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] - (18*n^2 + 63*n + 45 + (34*n^2 + 191*n + 202)*x + (16*n^2 + 192*n + 333)*x^2 + (64*n + 240)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 8*n + (8*n^2 - 3*n)*x + 8*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((10*n^3 + 9*n^2 - 34*n + 12 + (8*n^3 + 44*n^2 - 39*n - 8)*x + (32*n^2 + 28*n - 52)*x^2 + (32*n - 32)*x^3)*v[i - 1] - (18*n^2 + 63*n + 45 + (34*n^2 + 191*n + 202)*x + (16*n^2 + 192*n + 333)*x^2 + (64*n + 240)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 8*n + (8*n^2 - 3*n)*x + 8*n*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-e3d901814302c059","steps":["x \u21a6 x+1","D+1"],"edges":["e-31754865d296b758","e-a2e5d7c69a5239ae"],"text":"A128099 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A128099 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}