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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":"de2dcf4f11e5b708ef653431e0f742da3b52fe882ec0c0ace018d8f076347fd5","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:     1;\n  1:     0,       1;\n  2:    -3,      -1,      1;\n  3:     9,      -6,     -3,       1;\n  4:   -21,      43,     -4,      -6,     1;\n  5:    48,    -215,    104,      11,   -10,     1;\n  6:  -117,    1083,   -924,     145,    50,   -15,     1;\n  7:   285,   -6112,   7324,   -2491,    20,   127,   -21,   1;\n  8:  -273,   39731, -59975,   29421, -4506,  -581,   259, -28,   1;\n  9: -7092, -296133, 530100, -327840, 83524, -4067, -2212, 466, -36, 1;\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) = 1, P(1,x) = x, P(2,x) = -3 - x + x^2, P(3,x) = 9 - 6*x - 3*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[2, x_] := -3 - x + x^2;\nP[3, x_] := 9 - 6*x - 3*x^2 + x^3;\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] = 1; v[2] = x; v[3] = -3 - x + x^2; v[4] = 9 - 6*x - 3*x^2 + x^3; 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+1"],"edges":["e-6aed627db0c42e8b"],"text":"A137338 \u2192 D+1"},"headline":"A137338 \u2192 D+1","chain":null}