{"id":"n-bc6dfc9c53b05296","revision":0,"first_index":0,"fingerprint":"f6-9d29ecf489bb5c8f","name":"one_plus_derivative of basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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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":"11b9be6e2269f0d794525fab5a8b1944d044cefe9715c3afc4e745d8ead5c95f","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-7a2a38f140d4a893","role":"sequence","name":"basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:        4,        2;\n  2:       16,       18,        4;\n  3:       76,      124,       60,        8;\n  4:      428,      876,      620,      176,       16;\n  5:     2808,     6728,     6040,     2520,      480,       32;\n  6:    21096,    56888,    60376,    32200,     9040,     1248,      64;\n  7:   178848,   529408,   637616,   404992,   146160,    29792,    3136,    128;\n  8:  1690272,  5399424,  7188240,  5214944,  2254224,   593152,   92288,   7680,   256;\n  9: 17628480, 60029568, 86753280, 69960736, 34720896, 10991904, 2214912, 272640, 18432, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+4*n*x+4*x^2)*P(n-2,x) + (2*n+1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n*x + 4*x^2)*P[n - 2, x] + (2*n + 1 + 4*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n*x + 4*x^2)*v[i - 2] + (2*n + 1 + 4*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-60c7e36cbe887b7e","steps":["x \u21a6 x+1","D+1"],"edges":["e-01af55cf26557a61","e-3ab1f851d0f41b5b"],"text":"A137320 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A137320 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}