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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":"e1558d88ce35072179a7d7630994c451369b64a6664d195181ee54b349d6b59c","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) array.","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:  2,    1;\n  2:  7,    8,     1;\n  3: 12,   33,    14,     1;\n  4: 17,  108,    94,    20,     1;\n  5: 22,  233,   424,   190,    26,     1;\n  6: 27,  408,  1379,  1160,   321,    32,    1;\n  7: 32,  633,  3334,  5055,  2516,   487,   38,   1;\n  8: 37,  908,  6664, 16500, 13986,  4692,  688,  44,  1;\n  9: 42, 1233, 11744, 42620, 58356, 32022, 7888, 924, 50, 1;\n","recurrence_sentence":"The row polynomials satisfy -(5*n^3-13*n^2+8*n-(5*n^3-20*n^2+42*n)*x-(16*n^2-39*n)*x^2-(n^2+14*n)*x^3-n*x^4)*P(n,x) = -(5*n^3-8*n^2+3*n+(45*n^3-150*n^2+51*n)*x+(139*n^2-353*n+233)*x^2+(9*n^2+111*n-127)*x^3+(8*n-25)*x^4-x^5)*P(n-1,x) + ((5*n^2-3*n)*x+(90*n^2-37*n-8)*x^2+(5*n^2+304*n-127)*x^3+(35*n+299)*x^4+(n+35)*x^5+x^6)*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[(-(5*n^3 - 8*n^2 + 3*n + (45*n^3 - 150*n^2 + 51*n)*x + (139*n^2 - 353*n + 233)*x^2 + (9*n^2 + 111*n - 127)*x^3 + (8*n - 25)*x^4 - x^5)*P[n - 1, x] + ((5*n^2 - 3*n)*x + (90*n^2 - 37*n - 8)*x^2 + (5*n^2 + 304*n - 127)*x^3 + (35*n + 299)*x^4 + (n + 35)*x^5 + x^6)*D[P[n - 1, x], x])/(-5*n^3 + 13*n^2 - 8*n + (5*n^3 - 20*n^2 + 42*n)*x + (16*n^2 - 39*n)*x^2 + (n^2 + 14*n)*x^3 + n*x^4)]];\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] = (-(5*n^3 - 8*n^2 + 3*n + (45*n^3 - 150*n^2 + 51*n)*x + (139*n^2 - 353*n + 233)*x^2 + (9*n^2 + 111*n - 127)*x^3 + (8*n - 25)*x^4 - x^5)*v[i - 1] + ((5*n^2 - 3*n)*x + (90*n^2 - 37*n - 8)*x^2 + (5*n^2 + 304*n - 127)*x^3 + (35*n + 299)*x^4 + (n + 35)*x^5 + x^6)*deriv(v[i - 1], x))/(-5*n^3 + 13*n^2 - 8*n + (5*n^3 - 20*n^2 + 42*n)*x + (16*n^2 - 39*n)*x^2 + (n^2 + 14*n)*x^3 + n*x^4)); 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-ed9434bfd889c409","steps":["D+1"],"edges":["e-ca8ef6d6c750b1d1"],"text":"A081579 \u2192 D+1"},"headline":"A081579 \u2192 D+1","chain":null}