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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":"1979e4e61c09328e92d34fbb65218785093943afda7c35e8ad8f1388957ecf74","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-074c6703c2bee119","role":"sequence","name":"theta_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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:   2;\n  2:   8,     6;\n  3:  20,    60,     12;\n  4:  40,   300,    240,      20;\n  5:  70,  1050,   2100,     700,      30;\n  6: 112,  2940,  11760,    9800,    1680,      42;\n  7: 168,  7056,  49392,   82320,   35280,    3528,     56;\n  8: 240, 15120, 169344,  493920,  423360,  105840,   6720,    72;\n  9: 330, 29700, 498960, 2328480, 3492720, 1746360, 277200, 11880, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^3+6*n^2+5*n-12)*P(n,x) = (n^3+9*n^2+26*n+24+(10*n^3+15*n^2-25*n)*x)*P(n-1,x) + ((3*n^2+33*n+64)*x-(15*n^2+21*n-36)*x^2)*d/dx P(n-1,x) + ((6*n+14)*x^2+(6*n+14)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 + 15*n^2 - 25*n)*x)*P[n - 1, x] + ((3*n^2 + 33*n + 64)*x - (15*n^2 + 21*n - 36)*x^2)*D[P[n - 1, x], x] + ((6*n + 14)*x^2 + (6*n + 14)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 + 15*n^2 - 25*n)*x)*v[i - 1] + ((3*n^2 + 33*n + 64)*x - (15*n^2 + 21*n - 36)*x^2)*deriv(v[i - 1], x) + ((6*n + 14)*x^2 + (6*n + 14)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^3 + 6*n^2 + 5*n - 12)); 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-047228a38014e6d4","steps":["\u03b8+1","D"],"edges":["e-e126ec01ac61762b","e-cb9c2b27289527e6"],"text":"A056939 \u2192 \u03b8+1 \u2192 D"},"headline":"A056939 \u2192 \u03b8+1 \u2192 D","chain":null}