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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":"1369e547b2fe1c1b15235fff83f326eda727360748f564120f49a74fe8294695","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-c5818ec04781ce60","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= 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:           2,           1;\n  2:           5,           5,           1;\n  3:          27,          33,          11,          1;\n  4:         280,         374,         151,         22,         1;\n  5:        4860,        6818,        3038,        543,        40,        1;\n  6:      128060,      185288,       87635,      17559,      1618,       67,      1;\n  7:     4782420,     7067576,     3478497,     749625,     78738,     4158,    105,     1;\n  8:   240756400,   361308240,   182952704,   41465820,   4747413,   290208,   9506,   156,   1;\n  9: 15730936000, 23884778400, 12354394920, 2905494932, 353598138, 23850321, 916568, 19794, 222, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(2*n^2-8*n+10)*x+x^2)*P(n-2,x) + (2*n^2-6*n+7+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (2*n^2 - 8*n + 10)*x + x^2)*P[n - 2, x] + (2*n^2 - 6*n + 7 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (2*n^2 - 8*n + 10)*x + x^2)*v[i - 2] + (2*n^2 - 6*n + 7 + 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-54284d24a094e2d3","steps":["x \u21a6 x+1","D+1"],"edges":["e-0cfc062841fe3ed0","e-8f979227d9ebb990"],"text":"A269944 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A269944 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}