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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":"ca306a3e243438319aabc12d2ab0acecb83d63c5b782da368bf65f2cb32fa34b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bc0842751fa5b570","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:       6;\n  2:      45,        90;\n  3:     279,      1134,       1134;\n  4:    1638,     10080,      20412,      13608;\n  5:    9420,     77760,     238140,     320760,      160380;\n  6:   53703,    556470,    2285415,    4651020,     4691115,     1876446;\n  7:  305025,   3804570,   19615365,   53513460,    81486405,    65675610,    21891870;\n  8: 1730124,  25236288,  156648492,  536431896,  1094593500,  1331025696,   893188296,   255196656;\n  9: 9811314, 163863648, 1189767852, 4905371016, 12561953040, 20462131872, 20705728680, 11901444048, 2975361012;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2+3*n-6)*P(n,x) = (9*n^2+27*n+(18*n^2+54*n)*x)*P(n-1,x) + (4*n+18*n*x+18*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 + 27*n + (18*n^2 + 54*n)*x)*P[n - 1, x] + (4*n + 18*n*x + 18*n*x^2)*D[P[n - 1, x], x])/(3*n^2 + 3*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 + 27*n + (18*n^2 + 54*n)*x)*v[i - 1] + (4*n + 18*n*x + 18*n*x^2)*deriv(v[i - 1], x))/(3*n^2 + 3*n - 6)); 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-66e2c3448245b0b7","steps":["basis A013610","D"],"edges":["e-13974c051c6d411c","e-fa77d569c7ee9f38"],"text":"A033282 \u2192 basis A013610 \u2192 D"},"headline":"A033282 \u2192 basis A013610 \u2192 D","chain":null}