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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":"254b62223a846d99ad6fef78fdc3b7026caae08e3d8f8f179183ce26d6266a7b","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-07e5356ad37f61ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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:  3,    2;\n  3:  6,   12,     3;\n  4: 10,   42,    30,     4;\n  5: 15,  110,   165,    60,     5;\n  6: 21,  240,   645,   480,   105,     6;\n  7: 28,  462,  1995,  2660,  1155,   168,    7;\n  8: 36,  812,  5208, 11340,  8680,  2436,  252,   8;\n  9: 45, 1332, 11970, 39564, 49455, 23940, 4662, 360, 9;\n","recurrence_sentence":"The row polynomials satisfy -(n^2-n-(n^2+3*n-4)*x+(2*n-2)*x^2)*P(n,x) = -(n^2+n+(n^3-2*n^2-7*n)*x-(3*n^2-3*n)*x^2+2*n*x^3)*P(n-1,x) + ((2*n^2+2*n)*x^2-4*n*x^3)*d/dx 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[Together[(-(n^2 + n + (n^3 - 2*n^2 - 7*n)*x - (3*n^2 - 3*n)*x^2 + 2*n*x^3)*P[n - 1, x] + ((2*n^2 + 2*n)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (n^2 + 3*n - 4)*x - (2*n - 2)*x^2)]];\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] = (-(n^2 + n + (n^3 - 2*n^2 - 7*n)*x - (3*n^2 - 3*n)*x^2 + 2*n*x^3)*v[i - 1] + ((2*n^2 + 2*n)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (n^2 + 3*n - 4)*x - (2*n - 2)*x^2)); 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-07e5356ad37f61ea","steps":["D"],"edges":["e-0dc5896b91f11a29"],"text":"A100862 \u2192 D"},"headline":"A100862 \u2192 D","chain":null}