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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":"b343f90bdd80612c47d039ffa58a2889d84fa7edfc22d2e3885dd1b5cf4deeb4","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-7d332fcb7bf9a0f7","role":"sequence","name":"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:   1;\n  1:   1;\n  2:   3,   1;\n  3:   5,   2;\n  4:  10,   7,   1;\n  5:  18,  16,   3;\n  6:  33,  38,  12,  1;\n  7:  59,  82,  34,  4;\n  8: 105, 173,  93, 18, 1;\n  9: 185, 352, 231, 60, 5;\n","recurrence_sentence":"The row polynomials satisfy (n^3+2*n^2-4*n+(4*n^2-n)*x+4*n*x^2)*P(n,x) = (3*n^3+5*n^2-11*n+2+(2*n^3+14*n^2-9*n-4)*x+(8*n^2+10*n-14)*x^2+(8*n-8)*x^3)*P(n-1,x) - (5*n^2+20*n+15+(9*n^2+56*n+62)*x+(4*n^2+52*n+95)*x^2+(16*n+64)*x^3+16*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[((3*n^3 + 5*n^2 - 11*n + 2 + (2*n^3 + 14*n^2 - 9*n - 4)*x + (8*n^2 + 10*n - 14)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (5*n^2 + 20*n + 15 + (9*n^2 + 56*n + 62)*x + (4*n^2 + 52*n + 95)*x^2 + (16*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + 2*n^2 - 4*n + (4*n^2 - n)*x + 4*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] = ((3*n^3 + 5*n^2 - 11*n + 2 + (2*n^3 + 14*n^2 - 9*n - 4)*x + (8*n^2 + 10*n - 14)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (5*n^2 + 20*n + 15 + (9*n^2 + 56*n + 62)*x + (4*n^2 + 52*n + 95)*x^2 + (16*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + 2*n^2 - 4*n + (4*n^2 - n)*x + 4*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-75ccd0453a9b6cec","steps":["basis A007318","D+1"],"edges":["e-2a29a8b9938e2bd7","e-a0621b476dfe9301"],"text":"A011973 \u2192 basis A007318 \u2192 D+1"},"headline":"A011973 \u2192 basis A007318 \u2192 D+1","chain":null}