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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":"3ea500f68730cedbf5a01403aea5da388e8f8b907f5ff96f55388b5e1f3a52a5","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-8b333ce67e8457dc","role":"sequence","name":"basis_map_A013611 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:        0;\n  1:        8;\n  2:       56,       128;\n  3:      336,      1536,       1536;\n  4:     1936,     13312,      27648,      16384;\n  5:    11000,    101120,     322560,     409600,      163840;\n  6:    62184,    716160,    3095040,    6144000,     5406720,     1572864;\n  7:   351008,   4859904,   26557440,   72253440,   100925440,    66060288,    14680064;\n  8:  1981472,  32055296,  212029440,  735379456,  1422131200,  1497366528,   763363328,  134217728;\n  9: 11194344, 207226368, 1609961472, 6800080896, 16886661120, 24838668288, 20610809856, 8455716864, 1207959552;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+4*n-8)*P(n,x) = (28*n^2+4*n-8+(96*n^2-32*n-64)*x)*P(n-1,x) - (2*n+1+(24*n+12)*x+(64*n+32)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[((28*n^2 + 4*n - 8 + (96*n^2 - 32*n - 64)*x)*P[n - 1, x] - (2*n + 1 + (24*n + 12)*x + (64*n + 32)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 4*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((28*n^2 + 4*n - 8 + (96*n^2 - 32*n - 64)*x)*v[i - 1] - (2*n + 1 + (24*n + 12)*x + (64*n + 32)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 4*n - 8)); 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 A013611","D"],"edges":["e-034e0d2a0f7f14a7","e-44fee954becc1092"],"text":"A114687 \u2192 basis A013611 \u2192 D"},"headline":"A114687 \u2192 basis A013611 \u2192 D","chain":null}