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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":"987a9c9678e08f59ef1fe09a7695440cd81c6a7432728850fedd150a3fe94881","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-f46f66d0d087b029","role":"sequence","name":"Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).","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:      14,       10,         1;\n  1:      65,       81,        21,         1;\n  2:     294,      560,       264,        36,         1;\n  3:    1302,     3450,      2550,       650,        55,         1;\n  4:    5676,    19602,     20625,      8360,      1350,        78,        1;\n  5:   24453,   105105,    147147,     85995,     22295,      2499,      105,       1;\n  6:  104390,   539968,    958048,    754208,    284200,     51520,     4256,     136,      1;\n  7:  442442,  2685150,   5823792,   5877648,   3007368,    796824,   107100,    6804,    171,     1;\n  8: 1864356, 13016900,  33575850,  41860800,  27674640,  10011708,  1975050,  205200,  10350,   210,   1;\n  9: 7818538, 61829306, 185644250, 277880130, 228635550, 107469054, 29063958, 4446750, 368445, 15125, 253, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+26*n^3+66*n^2+78*n+36+(8*n^3+28*n^2+24*n)*x+(4*n^2+14*n+12)*x^2)*P(n,x) = (15*n^4+110*n^3+302*n^2+367*n+166+(12*n^4+109*n^3+270*n^2+232*n+49)*x+(24*n^3+97*n^2+131*n+60)*x^2+(12*n^2+35*n+25)*x^3)*P(n-1,x) + (n^3+5*n^2+9*n+6+(7*n^3+46*n^2+86*n+47)*x-(8*n^3+35*n^2+52*n+25)*x^2-(16*n^2+35*n+15)*x^3-(8*n+13)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 10*x + x^2.","mathematica":"P[0, x_] := 14 + 10*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n^4 + 110*n^3 + 302*n^2 + 367*n + 166 + (12*n^4 + 109*n^3 + 270*n^2 + 232*n + 49)*x + (24*n^3 + 97*n^2 + 131*n + 60)*x^2 + (12*n^2 + 35*n + 25)*x^3)*P[n - 1, x] + (n^3 + 5*n^2 + 9*n + 6 + (7*n^3 + 46*n^2 + 86*n + 47)*x - (8*n^3 + 35*n^2 + 52*n + 25)*x^2 - (16*n^2 + 35*n + 15)*x^3 - (8*n + 13)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 26*n^3 + 66*n^2 + 78*n + 36 + (8*n^3 + 28*n^2 + 24*n)*x + (4*n^2 + 14*n + 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 14 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^4 + 110*n^3 + 302*n^2 + 367*n + 166 + (12*n^4 + 109*n^3 + 270*n^2 + 232*n + 49)*x + (24*n^3 + 97*n^2 + 131*n + 60)*x^2 + (12*n^2 + 35*n + 25)*x^3)*v[i - 1] + (n^3 + 5*n^2 + 9*n + 6 + (7*n^3 + 46*n^2 + 86*n + 47)*x - (8*n^3 + 35*n^2 + 52*n + 25)*x^2 - (16*n^2 + 35*n + 15)*x^3 - (8*n + 13)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 26*n^3 + 66*n^2 + 78*n + 36 + (8*n^3 + 28*n^2 + 24*n)*x + (4*n^2 + 14*n + 12)*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-f46f66d0d087b029","steps":["D+1"],"edges":["e-14e7a5e7cec08aaa"],"text":"A110608 \u2192 D+1"},"headline":"A110608 \u2192 D+1","chain":null}