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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":"7f9348d6e384df0780c60d846bdff003b5e7e598ac05a5ffedd663653cdceafb","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:       0;\n  1:       6;\n  2:      42,        72;\n  3:     252,       864,       648;\n  4:    1452,      7488,     11664,       5184;\n  5:    8250,     56880,    136080,     129600,      38880;\n  6:   46638,    402840,   1305720,    1944000,    1283040,     279936;\n  7:  263256,   2733696,  11203920,   22861440,   23950080,   11757312,    1959552;\n  8: 1486104,  18031104,  89449920,  232678656,  337478400,  266499072,  101896704,  13436928;\n  9: 8395758, 116564832, 679202496, 2151588096, 4007283840, 4420749312, 2751211008, 846526464, 90699264;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2+3*n-6)*P(n,x) = (21*n^2+3*n-6+(54*n^2-18*n-36)*x)*P(n-1,x) - (2*n+1+(18*n+9)*x+(36*n+18)*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[((21*n^2 + 3*n - 6 + (54*n^2 - 18*n - 36)*x)*P[n - 1, x] - (2*n + 1 + (18*n + 9)*x + (36*n + 18)*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] = ((21*n^2 + 3*n - 6 + (54*n^2 - 18*n - 36)*x)*v[i - 1] - (2*n + 1 + (18*n + 9)*x + (36*n + 18)*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-8ce61d302cec295b","steps":["basis A013610","D"],"edges":["e-72b4d9a78d1bd227","e-f54ca35ed392967f"],"text":"A114687 \u2192 basis A013610 \u2192 D"},"headline":"A114687 \u2192 basis A013610 \u2192 D","chain":null}