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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":"7f08c31484119952804a4295ff99a35b1024cf5253f8be8ef1438731a795fcc8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b362ab32ef269db1","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      7,       3;\n   3:     32,      39,       9;\n   4:    122,     273,     171,       27;\n   5:    421,    1425,    1593,      675,       81;\n   6:   1369,    6300,   10503,     7830,     2511,      243;\n   7:   4283,   25092,   56511,    62262,    34830,     8991,     729;\n   8:  13043,   93111,  266301,   394335,   324000,   145071,   31347,    2187;\n   9:  38938,  328407, 1143873,  2145231,  2359530,  1544751,  576639,  107163,   6561;\n  10: 114484, 1115025, 4590099, 10470330, 14549868, 12711573, 6914565, 2213244, 360855, 19683;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3-243*n^2+465*n-333-(54*n-81)*x^2)*P(n,x) = (81*n^3-135*n^2+180*n-69+(81*n^3-162*n^2+90*n-9)*x-(162*n-135)*x^2-(81*n-81)*x^3)*P(n-1,x) + (45*n^2-90*n+50+(162*n^2-324*n+90)*x+(81*n^2-162*n-279)*x^2-324*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((81*n^3 - 135*n^2 + 180*n - 69 + (81*n^3 - 162*n^2 + 90*n - 9)*x - (162*n - 135)*x^2 - (81*n - 81)*x^3)*P[n - 1, x] + (45*n^2 - 90*n + 50 + (162*n^2 - 324*n + 90)*x + (81*n^2 - 162*n - 279)*x^2 - 324*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 - 243*n^2 + 465*n - 333 - (54*n - 81)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 - 135*n^2 + 180*n - 69 + (81*n^3 - 162*n^2 + 90*n - 9)*x - (162*n - 135)*x^2 - (81*n - 81)*x^3)*v[i - 1] + (45*n^2 - 90*n + 50 + (162*n^2 - 324*n + 90)*x + (81*n^2 - 162*n - 279)*x^2 - 324*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 - 243*n^2 + 465*n - 333 - (54*n - 81)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013610","D+1"],"edges":["e-258aa4ea5bc7ae92","e-90a2d750ed5f563f"],"text":"A111125 \u2192 basis A013610 \u2192 D+1"},"headline":"A111125 \u2192 basis A013610 \u2192 D+1","chain":null}