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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":"97ede6b6cfcd804f0dddacbae5770665c49bb11e53bb8af7f257a61d373dc3c2","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-3c983d83f25bade9","role":"sequence","name":"shift_x_plus_one of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 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:      6,       2;\n  2:     28,      20,       3;\n  3:    120,     136,      44,       4;\n  4:    495,     780,     405,      80,       5;\n  5:   2002,    4070,    2970,     950,     130,      6;\n  6:   8008,   20020,   19019,    8624,    1925,    196,     7;\n  7:  31824,   94640,  111384,   66248,   21112,   3528,   280,    8;\n  8: 125970,  434928,  612612,  454272,  191100,  45864,  6006,  384,   9;\n  9: 497420, 1957380, 3217080, 2870280, 1513680, 482160, 91140, 9660, 510, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^4+6*n^3+9*n^2+4*n+(2*n^3+8*n^2)*x+(n^2+4*n)*x^2)*P(n,x) = (4*n^4+22*n^3+44*n^2+38*n+12+(3*n^4+21*n^3+41*n^2+29*n+6)*x+(6*n^3+17*n^2+19*n+8)*x^2+(3*n^2+5*n+2)*x^3)*P(n-1,x) - ((2*n^3+9*n^2+13*n+6)*x+(2*n^3+13*n^2+21*n+10)*x^2+(4*n^2+10*n+6)*x^3+(2*n+2)*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[((4*n^4 + 22*n^3 + 44*n^2 + 38*n + 12 + (3*n^4 + 21*n^3 + 41*n^2 + 29*n + 6)*x + (6*n^3 + 17*n^2 + 19*n + 8)*x^2 + (3*n^2 + 5*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 + 9*n^2 + 13*n + 6)*x + (2*n^3 + 13*n^2 + 21*n + 10)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 9*n^2 + 4*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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] = ((4*n^4 + 22*n^3 + 44*n^2 + 38*n + 12 + (3*n^4 + 21*n^3 + 41*n^2 + 29*n + 6)*x + (6*n^3 + 17*n^2 + 19*n + 8)*x^2 + (3*n^2 + 5*n + 2)*x^3)*v[i - 1] - ((2*n^3 + 9*n^2 + 13*n + 6)*x + (2*n^3 + 13*n^2 + 21*n + 10)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 9*n^2 + 4*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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-0e6d847331141ca6","steps":["x \u21a6 x+1","D+1"],"edges":["e-1d39f03bcbd1a8b9","e-63840987c2c89c27"],"text":"A145596 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A145596 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}