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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":"6124937f8f900cc267fe6aa1756bcd782c3c9489aa3b462fef73262547adca53","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7a809bc95dc4396c","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          14,           1;\n  2:         195,          28,          1;\n  3:        2704,         585,         42,         1;\n  4:       37349,       10816,       1170,        56,        1;\n  5:      514098,      186745,      27040,      1950,       70,       1;\n  6:     7054567,     3084588,     560235,     54080,     2925,      84,      1;\n  7:    96536180,    49381969,   10796058,   1307215,    94640,    4095,     98,    1;\n  8:  1317718857,   772289440,  197527876,  28789488,  2614430,  151424,   5460,  112,   1;\n  9: 17946075862, 11859469713, 3475302480, 592583628, 64776348, 4705974, 227136, 7020, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+12+x)*P(n,x) = (13*n+169+(n+26)*x+x^2)*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[((13*n + 169 + (n + 26)*x + x^2)*P[n - 1, x])/(n + 12 + x)]];\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] = ((13*n + 169 + (n + 26)*x + x^2)*v[i - 1])/(n + 12 + x)); 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-81e6fb4f9f226d77","steps":["x \u21a6 x+1","D+1"],"edges":["e-4bd827707ee8c2a0","e-d061bcd70ae4099b"],"text":"A038327 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038327 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}