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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":"5338f80797c31f991c346c2281fa048426bb25d1c3de809f1f47326141621d86","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-a0e88202514b92bb","role":"sequence","name":"reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:  3,   2;\n  2:  6,  13,    4;\n  3:  9,  40,   40,     8;\n  4: 12,  85,  169,   108,    16;\n  5: 15, 148,  469,   582,   272,    32;\n  6: 18, 229, 1021,  2001,  1784,   656,    64;\n  7: 21, 328, 1906,  5256,  7328,  5072,  1536,   128;\n  8: 24, 445, 3205, 11562, 22601, 24248, 13680,  3520,  256;\n  9: 27, 580, 4999, 22458, 57431, 85994, 74640, 35488, 7936, 512;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3-6*n^2+3*n-(6*n^3-4*n^2+11*n)*x-(4*n^2-6*n)*x^2-(8*n^2-12*n)*x^3-8*n*x^4)*P(n,x) = -(3*n^3-3*n^2+(12*n^3-38*n^2-13*n)*x-(16*n^2+44*n-40)*x^2+(16*n^2-28*n+28)*x^3-16*x^4-16*x^5)*P(n-1,x) + (3*n^2*x+(24*n^2+4*n-3)*x^2+(12*n^2+36*n-20)*x^3+(48*n+24)*x^4+(16*n+48)*x^5+16*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[(-(3*n^3 - 3*n^2 + (12*n^3 - 38*n^2 - 13*n)*x - (16*n^2 + 44*n - 40)*x^2 + (16*n^2 - 28*n + 28)*x^3 - 16*x^4 - 16*x^5)*P[n - 1, x] + (3*n^2*x + (24*n^2 + 4*n - 3)*x^2 + (12*n^2 + 36*n - 20)*x^3 + (48*n + 24)*x^4 + (16*n + 48)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(-3*n^3 + 6*n^2 - 3*n + (6*n^3 - 4*n^2 + 11*n)*x + (4*n^2 - 6*n)*x^2 + (8*n^2 - 12*n)*x^3 + 8*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] = (-(3*n^3 - 3*n^2 + (12*n^3 - 38*n^2 - 13*n)*x - (16*n^2 + 44*n - 40)*x^2 + (16*n^2 - 28*n + 28)*x^3 - 16*x^4 - 16*x^5)*v[i - 1] + (3*n^2*x + (24*n^2 + 4*n - 3)*x^2 + (12*n^2 + 36*n - 20)*x^3 + (48*n + 24)*x^4 + (16*n + 48)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(-3*n^3 + 6*n^2 - 3*n + (6*n^3 - 4*n^2 + 11*n)*x + (4*n^2 - 6*n)*x^2 + (8*n^2 - 12*n)*x^3 + 8*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-d5c14b9fec875ecc","steps":["reverse","D+1"],"edges":["e-635d5bcbc56b7dbf","e-0cad163d732bb01e"],"text":"A121574 \u2192 reverse \u2192 D+1"},"headline":"A121574 \u2192 reverse \u2192 D+1","chain":null}