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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":"80541b097fc92f6b41d13fb26bd8a5412fc2380d5a86898f6c7718bf812275fd","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-7bf0ce499426260c","role":"sequence","name":"derivative 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:    0;\n  1:    2;\n  2:   14,     6;\n  3:   60,    64,     12;\n  4:  190,   390,    180,     20;\n  5:  490,  1680,   1470,    400,     30;\n  6: 1092,  5684,   8232,   4200,    770,     42;\n  7: 2184, 16128,  35280,  29568,  10080,   1344,    56;\n  8: 4020, 40068, 123984, 156240,  86940,  21420,  2184,   72;\n  9: 6930, 89760, 374220, 665280, 554400, 221760, 41580, 3360, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^4+n^3-6*n^2+20*n-16+(2*n^3+2*n^2-20*n+16)*x+(n^2+3*n-4)*x^2)*P(n,x) = (n^4+6*n^3+11*n^2+18*n+24+(3*n^4+2*n^3+8*n^2+4*n-32)*x+(6*n^3-10*n+14)*x^2+(3*n^2+4*n-2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2+4*n+7)*x-(2*n^3-2*n^2+2*n+13)*x^2-(4*n^2-9)*x^3-(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + 6*n^3 + 11*n^2 + 18*n + 24 + (3*n^4 + 2*n^3 + 8*n^2 + 4*n - 32)*x + (6*n^3 - 10*n + 14)*x^2 + (3*n^2 + 4*n - 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2 + 4*n + 7)*x - (2*n^3 - 2*n^2 + 2*n + 13)*x^2 - (4*n^2 - 9)*x^3 - (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^4 + n^3 - 6*n^2 + 20*n - 16 + (2*n^3 + 2*n^2 - 20*n + 16)*x + (n^2 + 3*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 6*n^3 + 11*n^2 + 18*n + 24 + (3*n^4 + 2*n^3 + 8*n^2 + 4*n - 32)*x + (6*n^3 - 10*n + 14)*x^2 + (3*n^2 + 4*n - 2)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2 + 4*n + 7)*x - (2*n^3 - 2*n^2 + 2*n + 13)*x^2 - (4*n^2 - 9)*x^3 - (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^4 + n^3 - 6*n^2 + 20*n - 16 + (2*n^3 + 2*n^2 - 20*n + 16)*x + (n^2 + 3*n - 4)*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":["D","D+1"],"edges":["e-e77b91d6fbb51ebc","e-18321cba73c57fa7"],"text":"A145596 \u2192 D \u2192 D+1"},"headline":"A145596 \u2192 D \u2192 D+1","chain":null}