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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":"31b5fec3d481284dfce4c7b824b52b61a319898390722198c3b9e0cfcc4b9604","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4ae73ef2b17633d9","role":"sequence","name":"Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:     -3,       2;\n  3:      6,     -12,      3;\n  4:     -9,      52,    -30,       4;\n  5:      3,    -218,    213,     -60,      5;\n  6:     81,    1002,  -1425,     620,   -105,      6;\n  7:   -786,   -5326,   9987,   -5820,   1475,   -168,    7;\n  8:   6711,   33020, -76485,   54916, -18235,   3066, -252,    8;\n  9: -60309, -235824, 648012, -543844, 219485, -47964, 5782, -360, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-3*n+2)*P(n-4,x) - (2*n^2-5*n+3-(2*n-2)*x)*P(n-3,x) - (n^2-(2*n-2)*x+x^2)*P(n-2,x) - (2*n-1-2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = -3 + 2*x, P(3,x) = 6 - 12*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := -3 + 2*x;\nP[3, x_] := 6 - 12*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 3*n + 2)*P[n - 4, x] - (2*n^2 - 5*n + 3 - (2*n - 2)*x)*P[n - 3, x] - (n^2 - (2*n - 2)*x + x^2)*P[n - 2, x] - (2*n - 1 - 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; v[3] = -3 + 2*x; v[4] = 6 - 12*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 3*n + 2)*v[i - 4] - (2*n^2 - 5*n + 3 - (2*n - 2)*x)*v[i - 3] - (n^2 - (2*n - 2)*x + x^2)*v[i - 2] - (2*n - 1 - 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-4ae73ef2b17633d9","steps":["D"],"edges":["e-c9e97b9a66627f15"],"text":"A137338 \u2192 D"},"headline":"A137338 \u2192 D","chain":null}