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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":"8a7913dfea31b852a45df8895de72127c0e17131514ea7d4affec90b201e1a40","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a18927332224e18f","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:     0;\n  2:     4;\n  3:    20,     32;\n  4:    76,    256,     192;\n  5:   260,   1344,    2112,    1024;\n  6:   840,   5856,   14208,   14336,    5120;\n  7:  2616,  22944,   75648,  117760,   87040,   24576;\n  8:  7940,  83904,  350400,  742400,  844800,  491520,  114688;\n  9: 23644, 292288, 1478976, 3978240, 6159360, 5505024, 2637824, 524288;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-28*n+24)*P(n,x) = (12*n^2-16*n+(16*n^2-16*n)*x)*P(n-1,x) + (5*n-5+(24*n-24)*x+(16*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 16*n + (16*n^2 - 16*n)*x)*P[n - 1, x] + (5*n - 5 + (24*n - 24)*x + (16*n - 16)*x^2)*D[P[n - 1, x], x])/(8*n^2 - 28*n + 24)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((12*n^2 - 16*n + (16*n^2 - 16*n)*x)*v[i - 1] + (5*n - 5 + (24*n - 24)*x + (16*n - 16)*x^2)*deriv(v[i - 1], x))/(8*n^2 - 28*n + 24)); 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-0eba401b117f16a9","steps":["basis A013611","D"],"edges":["e-39137b152517cefd","e-be2041da0226b806"],"text":"A085478 \u2192 basis A013611 \u2192 D"},"headline":"A085478 \u2192 basis A013611 \u2192 D","chain":null}