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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":"cb8486275179be8206e50bb26b3141c6ee3b7d963e61640389e739a36f6e3314","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-a36b2e24b7a3dbb7","role":"sequence","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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:       5,         6;\n  3:      26,        70,        45;\n  4:     154,       680,       945,        420;\n  5:    1044,      6608,     14700,      13860,       4725;\n  6:    8028,     67480,    211596,     315000,     225225,      62370;\n  7:   69264,    735768,   3026940,    6286280,    7033950,    4054050,     945945;\n  8:  663696,   8604432,  44332860,  119830480,  185735550,  166486320,   80405325,   16216200;\n  9: 6999840, 107922672, 673967448, 2267122000, 4547543000, 5617652040, 4199995800, 1745944200, 310134825;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+2*n^2*x+n^2*x^2)*P(n-2,x) - (n*x+2*n*x^2+n*x^3)*d/dx P(n-2,x) + (2*n+1+(2*n+2)*x)*P(n-1,x) + (x+x^2)*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[-(n^2 + 2*n^2*x + n^2*x^2)*P[n - 2, x] - (n*x + 2*n*x^2 + n*x^3)*D[P[n - 2, x], x] + (2*n + 1 + (2*n + 2)*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n^2*x + n^2*x^2)*v[i - 2] - (n*x + 2*n*x^2 + n*x^3)*deriv(v[i - 2], x) + (2*n + 1 + (2*n + 2)*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); 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-a36b2e24b7a3dbb7","steps":["D"],"edges":["e-158d9cf44f84435b"],"text":"A112486 \u2192 D"},"headline":"A112486 \u2192 D","chain":null}