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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":"75509b26f6f25471add2d576153db78ddc48a0e087829eaf1d769e125699e65a","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-8f767a3615460a6b","role":"sequence","name":"Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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:      3,       1;\n  2:     10,       7,       1;\n  3:     38,      43,      13,       1;\n  4:    161,     261,     133,      23,       1;\n  5:    747,    1606,    1203,     368,      41,       1;\n  6:   3753,   10158,   10253,    4728,     976,      75,      1;\n  7:  20253,   66529,   85475,   53973,   17216,    2553,    141,     1;\n  8: 116642,  452623,  711457,  578315,  256826,   60359,   6679,   271,   1;\n  9: 713130, 3202057, 5981815, 5997680, 3468608, 1154669, 207985, 17614, 529, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+1+2*x)*P(n,x) = -(n^3+2*n^2-n-2+(2*n^3+5*n^2-3*n-4)*x+(n^3+6*n^2-2*n-2)*x^2+(3*n^2+n-2)*x^3+(2*n-2)*x^4)*P(n-2,x) + (n^2+3*n+2+(3*n^2+11*n+8)*x+(3*n^2+16*n+14)*x^2+(n^2+11*n+14)*x^3+(3*n+8)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2+3*n+1+(2*n^2+6*n)*x+5*n*x^2+2*x^3)*P(n-1,x) - (n+1+(2*n+4)*x+(n+5)*x^2+2*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[(-(n^3 + 2*n^2 - n - 2 + (2*n^3 + 5*n^2 - 3*n - 4)*x + (n^3 + 6*n^2 - 2*n - 2)*x^2 + (3*n^2 + n - 2)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (n^2 + 3*n + 2 + (3*n^2 + 11*n + 8)*x + (3*n^2 + 16*n + 14)*x^2 + (n^2 + 11*n + 14)*x^3 + (3*n + 8)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 + 3*n + 1 + (2*n^2 + 6*n)*x + 5*n*x^2 + 2*x^3)*P[n - 1, x] - (n + 1 + (2*n + 4)*x + (n + 5)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*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] = (-(n^3 + 2*n^2 - n - 2 + (2*n^3 + 5*n^2 - 3*n - 4)*x + (n^3 + 6*n^2 - 2*n - 2)*x^2 + (3*n^2 + n - 2)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (n^2 + 3*n + 2 + (3*n^2 + 11*n + 8)*x + (3*n^2 + 16*n + 14)*x^2 + (n^2 + 11*n + 14)*x^3 + (3*n + 8)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 + 3*n + 1 + (2*n^2 + 6*n)*x + 5*n*x^2 + 2*x^3)*v[i - 1] - (n + 1 + (2*n + 4)*x + (n + 5)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","D+1"],"edges":["e-236d0fb15dc6eca0","e-5df37d25263414d4"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 D+1"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 D+1","chain":null}