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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":"05a056613315c8d80beb74f19337eab5c6381c86b334f854f7761edec207b99e","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-5939aa09e8ea2fc4","role":"sequence","name":"Coefficient triangle of certain polynomials N(5; m,x).","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:   7,    6;\n  2:  15,   56,    21;\n  3:  25,  192,   252,     56;\n  4:  37,  468,  1224,    840,     126;\n  5:  51,  950,  4050,   5400,    2310,     252;\n  6:  67, 1716, 10725,  23100,   18810,    5544,     462;\n  7:  85, 2856, 24486,  77000,  100485,   55440,   12012,    792;\n  8: 105, 4472, 50232, 216216,  410410,  360360,  144144,  24024,  1287;\n  9: 127, 6678, 95004, 535080, 1387386, 1765764, 1117116, 339768, 45045, 2002;\n","recurrence_sentence":"The row polynomials satisfy (n^4+8*n^3+12*n^2-15*n+(2*n^3+11*n^2+5*n)*x+(n^2+5*n)*x^2)*P(n,x) = (n^4+10*n^3+26*n^2+5*n+(3*n^4+26*n^3+68*n^2+65*n)*x+(6*n^3+33*n^2+71*n+40)*x^2+(3*n^2+13*n+20)*x^3)*P(n-1,x) + ((2*n^3+14*n^2+20*n)*x-(2*n^3+10*n^2+4*n-15)*x^2-(4*n^2+14*n+10)*x^3-(2*n+5)*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^4 + 10*n^3 + 26*n^2 + 5*n + (3*n^4 + 26*n^3 + 68*n^2 + 65*n)*x + (6*n^3 + 33*n^2 + 71*n + 40)*x^2 + (3*n^2 + 13*n + 20)*x^3)*P[n - 1, x] + ((2*n^3 + 14*n^2 + 20*n)*x - (2*n^3 + 10*n^2 + 4*n - 15)*x^2 - (4*n^2 + 14*n + 10)*x^3 - (2*n + 5)*x^4)*D[P[n - 1, x], x])/(n^4 + 8*n^3 + 12*n^2 - 15*n + (2*n^3 + 11*n^2 + 5*n)*x + (n^2 + 5*n)*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^4 + 10*n^3 + 26*n^2 + 5*n + (3*n^4 + 26*n^3 + 68*n^2 + 65*n)*x + (6*n^3 + 33*n^2 + 71*n + 40)*x^2 + (3*n^2 + 13*n + 20)*x^3)*v[i - 1] + ((2*n^3 + 14*n^2 + 20*n)*x - (2*n^3 + 10*n^2 + 4*n - 15)*x^2 - (4*n^2 + 14*n + 10)*x^3 - (2*n + 5)*x^4)*deriv(v[i - 1], x))/(n^4 + 8*n^3 + 12*n^2 - 15*n + (2*n^3 + 11*n^2 + 5*n)*x + (n^2 + 5*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-5939aa09e8ea2fc4","steps":["D+1"],"edges":["e-f9926095268ea913"],"text":"A062190 \u2192 D+1"},"headline":"A062190 \u2192 D+1","chain":null}