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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":"237ac640a48c451dcf7d3a4f1db7c8b146bf6c3392e5cacb89f7c4875fcfe015","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-b5ebaf3676d81658","role":"sequence","name":"basis_map_A013610 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:    1;\n  1:    1;\n  2:    6,    3;\n  3:   11,    6;\n  4:   32,   39,    9;\n  5:   69,  102,   27;\n  6:  166,  339,  189,   27;\n  7:  367,  894,  630,  108;\n  8:  828, 2439, 2349,  810,  81;\n  9: 1817, 6174, 7209, 3240, 405;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-9*n+(12*n^2-13*n)*x+12*n*x^2)*P(n,x) = (15*n^3-9*n^2-36*n+24+(18*n^3+42*n^2-101*n+40)*x+(72*n^2-18*n-42)*x^2+(72*n-72)*x^3)*P(n-1,x) - (18*n^2+36*n+12+(51*n^2+174*n+100)*x+(36*n^2+276*n+283)*x^2+(144*n+336)*x^3+144*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[((15*n^3 - 9*n^2 - 36*n + 24 + (18*n^3 + 42*n^2 - 101*n + 40)*x + (72*n^2 - 18*n - 42)*x^2 + (72*n - 72)*x^3)*P[n - 1, x] - (18*n^2 + 36*n + 12 + (51*n^2 + 174*n + 100)*x + (36*n^2 + 276*n + 283)*x^2 + (144*n + 336)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 - 9*n + (12*n^2 - 13*n)*x + 12*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] = ((15*n^3 - 9*n^2 - 36*n + 24 + (18*n^3 + 42*n^2 - 101*n + 40)*x + (72*n^2 - 18*n - 42)*x^2 + (72*n - 72)*x^3)*v[i - 1] - (18*n^2 + 36*n + 12 + (51*n^2 + 174*n + 100)*x + (36*n^2 + 276*n + 283)*x^2 + (144*n + 336)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 - 9*n + (12*n^2 - 13*n)*x + 12*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-7d332fcb7bf9a0f7","steps":["basis A013610","D+1"],"edges":["e-95533c5935436804","e-acf97d7fab462139"],"text":"A128100 \u2192 basis A013610 \u2192 D+1"},"headline":"A128100 \u2192 basis A013610 \u2192 D+1","chain":null}