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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":"9b63ae36ca60aca18ea2fd1cd3b2bb826f87118c725fe466d5db59309ad187f9","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-b0d4d18dfd70fd8a","role":"sequence","name":"basis_map_A013610 of Even row Pascal-square read by antidiagonals.","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:      9,       6;\n  2:     38,      84,      27;\n  3:    135,     510,     522,      108;\n  4:    445,    2388,    4239,     2700,      405;\n  5:   1404,    9792,   25002,    27972,    12555,     1458;\n  6:   4301,   36972,  123768,   201312,   160461,    54432,     5103;\n  7:  12897,  131898,  547614,  1176660,  1376190,   836892,   224532,    17496;\n  8:  38050,  451428, 2240109,  6002316,  9335655,  8398080,  4072194,   892296,   59049;\n  9: 110835, 1496874, 8647362, 27807300, 54196695, 65218770, 47116728, 18790704, 3444525, 196830;\n","recurrence_sentence":"The row polynomials satisfy (216*n^3-144*n^2-60*n+(432*n^2-378*n)*x+216*n*x^2)*P(n,x) = (864*n^3-144*n^2-690*n+78+(1944*n^3+1080*n^2-2862*n+396)*x+(3888*n^2-1890*n+270)*x^2+(1944*n-648)*x^3)*P(n-1,x) - (180*n^2+150*n-5+(972*n^2+1170*n+18)*x+(1296*n^2+3024*n+387)*x^2+(2592*n+1296)*x^3+1296*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[((864*n^3 - 144*n^2 - 690*n + 78 + (1944*n^3 + 1080*n^2 - 2862*n + 396)*x + (3888*n^2 - 1890*n + 270)*x^2 + (1944*n - 648)*x^3)*P[n - 1, x] - (180*n^2 + 150*n - 5 + (972*n^2 + 1170*n + 18)*x + (1296*n^2 + 3024*n + 387)*x^2 + (2592*n + 1296)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(216*n^3 - 144*n^2 - 60*n + (432*n^2 - 378*n)*x + 216*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] = ((864*n^3 - 144*n^2 - 690*n + 78 + (1944*n^3 + 1080*n^2 - 2862*n + 396)*x + (3888*n^2 - 1890*n + 270)*x^2 + (1944*n - 648)*x^3)*v[i - 1] - (180*n^2 + 150*n - 5 + (972*n^2 + 1170*n + 18)*x + (1296*n^2 + 3024*n + 387)*x^2 + (2592*n + 1296)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(216*n^3 - 144*n^2 - 60*n + (432*n^2 - 378*n)*x + 216*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-79ebafb266da3b41","steps":["basis A013610","D+1"],"edges":["e-01dc3c5cce6c6f01","e-fed3ca2b3adbdcd3"],"text":"A172431 \u2192 basis A013610 \u2192 D+1"},"headline":"A172431 \u2192 basis A013610 \u2192 D+1","chain":null}