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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":"5cbb8a353eefc9ba841b657d1746d21832e5a1acf32c21bc00ae13da743500f4","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-0c5ba5f813be2bea","role":"sequence","name":"shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.","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:        3,        1;\n  2:       12,        8,         1;\n  3:       62,       70,        19,         1;\n  4:      372,      624,       316,        48,        1;\n  5:     2492,     5636,      4368,      1328,      131,        1;\n  6:    18384,    52768,     56028,     26688,     5456,      376,       1;\n  7:   147800,   517336,    705500,    468548,   153926,    22398,    1107,      1;\n  8:  1283088,  5321088,   8926048,   7710720,  3602056,   869216,   93192,   3296,    1;\n  9: 11939376, 57393040, 114784768, 123034880, 75727928, 26533064, 4895920, 396784, 9859, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+1+4*x)*P(n,x) = -(8*n^3-4*n^2-20*n+6+(16*n^3+8*n^2-68*n+26)*x+(8*n^3+36*n^2-78*n+21)*x^2+(24*n^2-16*n-22)*x^3+(16*n-24)*x^4)*P(n-2,x) + (8*n^2+12*n+(24*n^2+56*n+6)*x+(24*n^2+100*n+32)*x^2+(8*n^2+80*n+62)*x^3+(24*n+52)*x^4+16*x^5)*d/dx P(n-2,x) + (8*n^2-4+(8*n^2+16*n-18)*x+(20*n-10)*x^2+8*x^3)*P(n-1,x) - (4*n+2+(8*n+12)*x+(4*n+18)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 4*n^2 - 20*n + 6 + (16*n^3 + 8*n^2 - 68*n + 26)*x + (8*n^3 + 36*n^2 - 78*n + 21)*x^2 + (24*n^2 - 16*n - 22)*x^3 + (16*n - 24)*x^4)*P[n - 2, x] + (8*n^2 + 12*n + (24*n^2 + 56*n + 6)*x + (24*n^2 + 100*n + 32)*x^2 + (8*n^2 + 80*n + 62)*x^3 + (24*n + 52)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (8*n^2 - 4 + (8*n^2 + 16*n - 18)*x + (20*n - 10)*x^2 + 8*x^3)*P[n - 1, x] - (4*n + 2 + (8*n + 12)*x + (4*n + 18)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(2*n + 1 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 4*n^2 - 20*n + 6 + (16*n^3 + 8*n^2 - 68*n + 26)*x + (8*n^3 + 36*n^2 - 78*n + 21)*x^2 + (24*n^2 - 16*n - 22)*x^3 + (16*n - 24)*x^4)*v[i - 2] + (8*n^2 + 12*n + (24*n^2 + 56*n + 6)*x + (24*n^2 + 100*n + 32)*x^2 + (8*n^2 + 80*n + 62)*x^3 + (24*n + 52)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (8*n^2 - 4 + (8*n^2 + 16*n - 18)*x + (20*n - 10)*x^2 + 8*x^3)*v[i - 1] - (4*n + 2 + (8*n + 12)*x + (4*n + 18)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(2*n + 1 + 4*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-28595a60eb57b37f","steps":["x \u21a6 x+1","D+1"],"edges":["e-4c7248607accd612","e-1342e1700dd7e20d"],"text":"A039756 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A039756 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}