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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":"107765ebaf7c9f760b4e56283efc2b4a874e33710660111e14565a4f7aa38a5b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3897d3bccfbe11da","role":"sequence","name":"basis_map_A013609 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","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:      7,       4;\n  3:     34,      48,      12;\n  4:    155,     360,     216,      32;\n  5:    686,    2240,    2280,     800,      80;\n  6:   2982,   12600,   18480,   11200,    2640,     192;\n  7:  12804,   66528,  128520,  114240,   47040,    8064,    448;\n  8:  54483,  336336,  809424,  960960,  588000,  177408,  23296,  1024;\n  9: 230230, 1647360, 4756752, 7111104, 5876640, 2661120, 618240, 64512, 2304;\n","recurrence_sentence":"The row polynomials satisfy (2*n^4-6*n^3+5*n^2-n+(4*n^3-11*n^2+7*n)*x+(2*n^2-2*n)*x^2)*P(n,x) = (8*n^4-20*n^3+8*n^2+10*n-6+(12*n^4-20*n^3-24*n^2+54*n-22)*x+(24*n^3-62*n^2+54*n-16)*x^2+(12*n^2-20*n+8)*x^3)*P(n-1,x) - ((4*n^3-4*n^2-2*n+2)*x+(8*n^3-14*n+7)*x^2+(16*n^2-16*n+4)*x^3+(8*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^4 - 20*n^3 + 8*n^2 + 10*n - 6 + (12*n^4 - 20*n^3 - 24*n^2 + 54*n - 22)*x + (24*n^3 - 62*n^2 + 54*n - 16)*x^2 + (12*n^2 - 20*n + 8)*x^3)*P[n - 1, x] - ((4*n^3 - 4*n^2 - 2*n + 2)*x + (8*n^3 - 14*n + 7)*x^2 + (16*n^2 - 16*n + 4)*x^3 + (8*n - 4)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 6*n^3 + 5*n^2 - n + (4*n^3 - 11*n^2 + 7*n)*x + (2*n^2 - 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^4 - 20*n^3 + 8*n^2 + 10*n - 6 + (12*n^4 - 20*n^3 - 24*n^2 + 54*n - 22)*x + (24*n^3 - 62*n^2 + 54*n - 16)*x^2 + (12*n^2 - 20*n + 8)*x^3)*v[i - 1] - ((4*n^3 - 4*n^2 - 2*n + 2)*x + (8*n^3 - 14*n + 7)*x^2 + (16*n^2 - 16*n + 4)*x^3 + (8*n - 4)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 6*n^3 + 5*n^2 - n + (4*n^3 - 11*n^2 + 7*n)*x + (2*n^2 - 2*n)*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-b1b19325e0094118","steps":["basis A013609","D+1"],"edges":["e-169cfb9127a1a777","e-40233cc6eaa43cf6"],"text":"A367270 \u2192 basis A013609 \u2192 D+1"},"headline":"A367270 \u2192 basis A013609 \u2192 D+1","chain":null}