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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":"f8c9d66be0e351e8958ad1cc0f95a098f8d67e11af979adce0b77aa04a2ea36d","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-e655e253465c7aff","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.","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,   3,   1;\n  1:  3,   4,   1;\n  2:  1,   7,  12,   7,    1;\n  3:  5,  16,  18,   8,    1;\n  4:  1,  11,  35,  50,   35,   11,    1;\n  5:  7,  36,  75,  80,   45,   12,    1;\n  6:  1,  15,  70, 161,  210,  161,   70,  15,   1;\n  7:  9,  64, 196, 336,  350,  224,   84,  16,   1;\n  8:  1,  19, 117, 372,  714,  882,  714, 372, 117, 19, 1;\n  9: 11, 100, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n+1+(2*n+2)*x+x^2)*P(n,x) = -(n^3+2*n^2-2*n-1-(n^2+7*n+5)*x-(n^3+5*n^2+10*n+8)*x^2-(2*n^2+6*n+5)*x^3-(n+1)*x^4)*P(n-1,x) + (n^2+3*n+1+(n^2+3*n-1)*x-(n^2+5*n+9)*x^2-(n^2+7*n+12)*x^3-(2*n+6)*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x + x^2.","mathematica":"P[0, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - 2*n - 1 - (n^2 + 7*n + 5)*x - (n^3 + 5*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 5)*x^3 - (n + 1)*x^4)*P[n - 1, x] + (n^2 + 3*n + 1 + (n^2 + 3*n - 1)*x - (n^2 + 5*n + 9)*x^2 - (n^2 + 7*n + 12)*x^3 - (2*n + 6)*x^4 - x^5)*D[P[n - 1, x], x])/(n^2 + 3*n + 1 + (2*n + 2)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 3*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - 2*n - 1 - (n^2 + 7*n + 5)*x - (n^3 + 5*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 5)*x^3 - (n + 1)*x^4)*v[i - 1] + (n^2 + 3*n + 1 + (n^2 + 3*n - 1)*x - (n^2 + 5*n + 9)*x^2 - (n^2 + 7*n + 12)*x^3 - (2*n + 6)*x^4 - x^5)*deriv(v[i - 1], x))/(n^2 + 3*n + 1 + (2*n + 2)*x + 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-e655e253465c7aff","steps":["D+1"],"edges":["e-6d9afb48cd1e25d2"],"text":"Veronese parent of A109447 \u2192 D+1"},"headline":"Veronese parent of A109447 \u2192 D+1","chain":null}