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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":"cc1d30835ba0c588b89af01db0b238d20d5e451f55d08f8b5823ceaca350e974","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-4172472abe953cea","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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:          0;\n  1:          1;\n  2:          6,          2;\n  3:         59,         30,          3;\n  4:        812,        508,         84,         4;\n  5:      14389,      10260,       2190,       180,        5;\n  6:     312114,     244238,      61380,      6700,      330,       6;\n  7:    8011695,    6730178,    1900857,    249340,    16625,     546,     7;\n  8:  237560280,  211198552,   65220120,   9765336,   793800,   35784,   840,    8;\n  9: 7990901865, 7444672776, 2469061788, 409216248, 38402070, 2133432, 69468, 1224, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-56*n+49+(8*n-14)*x+x^2)*P(n-2,x) + (8*n-10+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 56*n + 49 + (8*n - 14)*x + x^2)*P[n - 2, x] + (8*n - 10 + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 56*n + 49 + (8*n - 14)*x + x^2)*v[i - 2] + (8*n - 10 + 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-4172472abe953cea","steps":["D"],"edges":["e-29e3872091fc0ea2"],"text":"A290319 \u2192 D"},"headline":"A290319 \u2192 D","chain":null}