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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":"954f47bcc2942cae8432309c8777d6a0c17522770e2861d696ce702ab9064226","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-1a0f876fa1441737","role":"sequence","name":"basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. 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:         1;\n  1:         1,         1;\n  2:         3,         5,         1;\n  3:        18,        36,        12,         1;\n  4:       162,       360,       153,        22,        1;\n  5:      1944,      4644,      2295,       435,       35,       1;\n  6:     29160,     73548,     40419,      9135,      990,      51,      1;\n  7:    524880,   1382184,    823284,    210924,    27720,    1953,     70,    1;\n  8:  11022480,  30075624,  19099800,   5384232,   812889,   70308,   3486,   92,   1;\n  9: 264539520, 743859936, 497997396, 151512444, 25441857, 2553201, 156870, 5778, 117, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-36*n+36+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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-1a0f876fa1441737","steps":["D+1"],"edges":["e-e10ad6167182ea44"],"text":"basis-map coefficients of A371080, shifted \u2192 D+1"},"headline":"basis-map coefficients of A371080, shifted \u2192 D+1","chain":null}