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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":"76f56d20dbc523b8c3e1099ed3837e98f093d382f3003c69f9081b5ec5a9c297","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-3b748f1f6d80b05e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:       9,         6;\n  2:      53,       114,        36;\n  3:     297,      1116,      1080,        216;\n  4:    1649,      8940,     15408,       9072,       1296;\n  5:    9153,     65130,    164520,     174960,      71280,       7776;\n  6:   50917,    449478,   1507140,    2379240,    1769040,     536544,      46656;\n  7:  284049,   2996952,  12570768,   26596080,   29665440,   16547328,    3919104,     279936;\n  8: 1589153,  19517208,  98465472,  262495296,  395648064,  333994752,  146313216,   27993600,   1679616;\n  9: 8914617, 124960590, 737484912, 2377988928, 4545180864, 5222206080, 3488002560, 1239556608, 196515072, 10077696;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3+90*n^2-51*n-30+(108*n^2+135*n-162)*x+(54*n+108)*x^2)*P(n,x) = (378*n^3+306*n^2-87*n-30+(972*n^3+1080*n^2-783*n-162)*x+(1944*n^2-108*n+108)*x^2+972*n*x^3)*P(n-1,x) - (36*n^2+42*n+8+(324*n^2+450*n+90)*x+(648*n^2+1404*n+342)*x^2+(1296*n+648)*x^3+648*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[((378*n^3 + 306*n^2 - 87*n - 30 + (972*n^3 + 1080*n^2 - 783*n - 162)*x + (1944*n^2 - 108*n + 108)*x^2 + 972*n*x^3)*P[n - 1, x] - (36*n^2 + 42*n + 8 + (324*n^2 + 450*n + 90)*x + (648*n^2 + 1404*n + 342)*x^2 + (1296*n + 648)*x^3 + 648*x^4)*D[P[n - 1, x], x])/(54*n^3 + 90*n^2 - 51*n - 30 + (108*n^2 + 135*n - 162)*x + (54*n + 108)*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] = ((378*n^3 + 306*n^2 - 87*n - 30 + (972*n^3 + 1080*n^2 - 783*n - 162)*x + (1944*n^2 - 108*n + 108)*x^2 + 972*n*x^3)*v[i - 1] - (36*n^2 + 42*n + 8 + (324*n^2 + 450*n + 90)*x + (648*n^2 + 1404*n + 342)*x^2 + (1296*n + 648)*x^3 + 648*x^4)*deriv(v[i - 1], x))/(54*n^3 + 90*n^2 - 51*n - 30 + (108*n^2 + 135*n - 162)*x + (54*n + 108)*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-8ce61d302cec295b","steps":["basis A013610","D+1"],"edges":["e-72b4d9a78d1bd227","e-e798f2cd39afc45f"],"text":"A114687 \u2192 basis A013610 \u2192 D+1"},"headline":"A114687 \u2192 basis A013610 \u2192 D+1","chain":null}