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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":"ec4069f4f19fa4e66e869b1a00e2275cdff60bd12ff1a8a5c35b03da32d255ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6a6677e3386580cd","role":"sequence","name":"basis_map_A013609 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph 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:     1;\n  2:     4,     2;\n  3:    10,     6;\n  4:    34,    48,     12;\n  5:   106,   200,     60;\n  6:   376,  1020,    720,    120;\n  7:  1324,  4452,   4200,    840;\n  8:  5020, 21056,  28560,  13440,  1680;\n  9: 19324, 95328, 160272, 100800, 15120;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-1+(4*n-6)*x+4*x^2)*P(n,x) = (n^3-n^2+n-2+(2*n^3-2*n-2)*x+(8*n^2-8*n+4)*x^2+(8*n-8)*x^3)*P(n-1,x) - (n^2+n+1+(4*n^2+8*n+6)*x+(4*n^2+20*n+16)*x^2+(16*n+24)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 + n - 2 + (2*n^3 - 2*n - 2)*x + (8*n^2 - 8*n + 4)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (n^2 + n + 1 + (4*n^2 + 8*n + 6)*x + (4*n^2 + 20*n + 16)*x^2 + (16*n + 24)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^2 - n - 1 + (4*n - 6)*x + 4*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 - n^2 + n - 2 + (2*n^3 - 2*n - 2)*x + (8*n^2 - 8*n + 4)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (n^2 + n + 1 + (4*n^2 + 8*n + 6)*x + (4*n^2 + 20*n + 16)*x^2 + (16*n + 24)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^2 - n - 1 + (4*n - 6)*x + 4*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-97b51734d39ede6c","steps":["basis A013609","D+1"],"edges":["e-4139d19ff9e22325","e-997a8920e39492f4"],"text":"A100861 \u2192 basis A013609 \u2192 D+1"},"headline":"A100861 \u2192 basis A013609 \u2192 D+1","chain":null}