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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":"57d16f6f352481352917ebd2ef967d9cd0a52b0b3f777bb6f3aa4895434f9564","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-98aa86c8bf356480","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:      33,       36;\n  1:     132,      306,      162;\n  2:     465,     1674,     1863,       648;\n  3:    1530,     7506,    12960,      9396,      2430;\n  4:    4824,    30024,    70794,     79380,     42525,      8748;\n  5:   14772,   111510,   334530,    511920,    422820,    179334,     30618;\n  6:   44283,   393138,  1434591,   2794500,   3145635,   2051406,    719523,    104976;\n  7:  130620,  1333602,  5738364,  13609944,  19493460,  17299170,   9307872,   2781864,    354294;\n  8:  380415,  4391388, 21781710,  60988788, 106372035, 120022560,  87751188,  40153320,  10451673,  1180980;\n  9: 1096596, 14123052, 79368012, 256483260, 528330600, 725707836, 674024652, 418539312, 166518180, 38381850, 3897234;\n","recurrence_sentence":"The row polynomials satisfy (6*n^2-12*n-18+(36*n^2+90*n+54)*x)*P(n,x) = (9*n^2-6*n-99+(63*n^2+252*n+189)*x+(54*n^2+270*n+324)*x^2)*P(n-1,x) + (5*n-5+(48*n+42)*x+(117*n+207)*x^2+(54*n+108)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 33 + 36*x.","mathematica":"P[0, x_] := 33 + 36*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 6*n - 99 + (63*n^2 + 252*n + 189)*x + (54*n^2 + 270*n + 324)*x^2)*P[n - 1, x] + (5*n - 5 + (48*n + 42)*x + (117*n + 207)*x^2 + (54*n + 108)*x^3)*D[P[n - 1, x], x])/(6*n^2 - 12*n - 18 + (36*n^2 + 90*n + 54)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 33 + 36*x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 - 6*n - 99 + (63*n^2 + 252*n + 189)*x + (54*n^2 + 270*n + 324)*x^2)*v[i - 1] + (5*n - 5 + (48*n + 42)*x + (117*n + 207)*x^2 + (54*n + 108)*x^3)*deriv(v[i - 1], x))/(6*n^2 - 12*n - 18 + (36*n^2 + 90*n + 54)*x)); 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-dfab35176bd08e72","steps":["basis A013610","D"],"edges":["e-77ee9774535f66a4","e-d4405beb16e9f37c"],"text":"A207606 \u2192 basis A013610 \u2192 D"},"headline":"A207606 \u2192 basis A013610 \u2192 D","chain":null}