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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":"339b0a026692414c10c333970637c770fb455e34c2db3f2ae573bbc1202a0733","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-261c6497576cc5f7","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:          1;\n  2:         10,          6;\n  3:        100,        136,          45;\n  4:       1136,       2488,        1788,         420;\n  5:      14816,      45328,       51468,       25656,        4725;\n  6:     220032,     869888,     1365528,     1062736,      409470,       62370;\n  7:    3679488,   17899264,    36080208,    38547456,    23002040,     7260912,      945945;\n  8:   68548608,  396772864,   980016768,  1338363456,  1090797440,   530211312,   142201080,   16216200;\n  9: 1409347584, 9472564224, 27758009856, 46304255616, 48074577360, 31796071392, 13075477128, 3054655440, 310134825;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(16*n^3-48*n^2+44*n-12)*x+(8*n^3-28*n^2+30*n-9)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(8*n^2-16*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1])/(2*n - 3)); 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-2b477f2aa77e3efc","steps":["basis A007318","D"],"edges":["e-07dfe9939e4be29b","e-b30f2b3de0b37d8c"],"text":"A109692 \u2192 basis A007318 \u2192 D"},"headline":"A109692 \u2192 basis A007318 \u2192 D","chain":null}