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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":"6693f9816aafdcb8cd97da55a844c8616cf4bcfe14f323a120b4dee7a256b476","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-b6cda967a092feb1","role":"sequence","name":"basis_map_A007318 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:       5,        2;\n  2:      25,       22,        4;\n  3:     129,      180,       72,        8;\n  4:     681,     1316,      848,      208,       16;\n  5:    3653,     9070,     8200,     3280,      560,      32;\n  6:   19825,    60306,    70740,    40120,    11280,    1440,      64;\n  7:  108545,   391496,   566832,   420560,   169120,   35840,    3584,    128;\n  8:  598417,  2498824,  4314688,  3976896,  2106944,  643328,  107520,   8704,   256;\n  9: 3317445, 15750234, 31631472, 34948032, 23131584, 9362304, 2268672, 308736, 20736, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+6*n^2+3*n-2+(4*n^2+7*n-2)*x+(2*n+4)*x^2)*P(n,x) = (14*n^3+30*n^2+3*n-2+(12*n^3+48*n^2+5*n-2)*x+(24*n^2+20*n+4)*x^2+12*n*x^3)*P(n-1,x) - (4*n^2+10*n+4+(12*n^2+38*n+18)*x+(8*n^2+44*n+30)*x^2+(16*n+24)*x^3+8*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[((14*n^3 + 30*n^2 + 3*n - 2 + (12*n^3 + 48*n^2 + 5*n - 2)*x + (24*n^2 + 20*n + 4)*x^2 + 12*n*x^3)*P[n - 1, x] - (4*n^2 + 10*n + 4 + (12*n^2 + 38*n + 18)*x + (8*n^2 + 44*n + 30)*x^2 + (16*n + 24)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*n^2 + 3*n - 2 + (4*n^2 + 7*n - 2)*x + (2*n + 4)*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] = ((14*n^3 + 30*n^2 + 3*n - 2 + (12*n^3 + 48*n^2 + 5*n - 2)*x + (24*n^2 + 20*n + 4)*x^2 + 12*n*x^3)*v[i - 1] - (4*n^2 + 10*n + 4 + (12*n^2 + 38*n + 18)*x + (8*n^2 + 44*n + 30)*x^2 + (16*n + 24)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*n^2 + 3*n - 2 + (4*n^2 + 7*n - 2)*x + (2*n + 4)*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 A007318","D+1"],"edges":["e-6804282af6df94ec","e-7a3305a07c5fc71a"],"text":"A114687 \u2192 basis A007318 \u2192 D+1"},"headline":"A114687 \u2192 basis A007318 \u2192 D+1","chain":null}