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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":"0825ce5d481a631b11462e113b56471edf1ae27f080e4788e226b919b06c3bd8","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-0bad942bfffd4f03","role":"sequence","name":"basis_map_A007318 of Pascal-(1,4,1) array.","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:     16,     10,      1;\n  3:     60,     64,     17,      1;\n  4:    240,    360,    160,     24,      1;\n  5:    896,   1760,   1160,    304,     31,      1;\n  6:   3328,   8096,   7120,   2784,    496,     38,     1;\n  7:  12096,  35200,  39056,  20784,   5556,    736,    45,    1;\n  8:  43520, 147584, 198272, 134720,  49376,   9800,  1024,   52,  1;\n  9: 154880, 600320, 949120, 789248, 372752, 101680, 15840, 1360, 59, 1;\n","recurrence_sentence":"The row polynomials satisfy (10*n^2+10*n+(5*n^3+15*n^2+10*n)*x+(19*n^2+9*n)*x^2+(n^2+18*n)*x^3+n*x^4)*P(n,x) = -(50*n^3-10*n^2-180*n+80+(45*n^3+155*n^2-290*n-20)*x+(166*n^2+28*n-308)*x^2+(9*n^2+143*n-237)*x^3+(8*n-30)*x^4-x^5)*P(n-1,x) + (100*n^2+300*n+200+(200*n^2+980*n+980)*x+(105*n^2+1095*n+1770)*x^2+(5*n^2+454*n+1439)*x^3+(40*n+489)*x^4+(n+41)*x^5+x^6)*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[(-(50*n^3 - 10*n^2 - 180*n + 80 + (45*n^3 + 155*n^2 - 290*n - 20)*x + (166*n^2 + 28*n - 308)*x^2 + (9*n^2 + 143*n - 237)*x^3 + (8*n - 30)*x^4 - x^5)*P[n - 1, x] + (100*n^2 + 300*n + 200 + (200*n^2 + 980*n + 980)*x + (105*n^2 + 1095*n + 1770)*x^2 + (5*n^2 + 454*n + 1439)*x^3 + (40*n + 489)*x^4 + (n + 41)*x^5 + x^6)*D[P[n - 1, x], x])/(10*n^2 + 10*n + (5*n^3 + 15*n^2 + 10*n)*x + (19*n^2 + 9*n)*x^2 + (n^2 + 18*n)*x^3 + n*x^4)]];\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] = (-(50*n^3 - 10*n^2 - 180*n + 80 + (45*n^3 + 155*n^2 - 290*n - 20)*x + (166*n^2 + 28*n - 308)*x^2 + (9*n^2 + 143*n - 237)*x^3 + (8*n - 30)*x^4 - x^5)*v[i - 1] + (100*n^2 + 300*n + 200 + (200*n^2 + 980*n + 980)*x + (105*n^2 + 1095*n + 1770)*x^2 + (5*n^2 + 454*n + 1439)*x^3 + (40*n + 489)*x^4 + (n + 41)*x^5 + x^6)*deriv(v[i - 1], x))/(10*n^2 + 10*n + (5*n^3 + 15*n^2 + 10*n)*x + (19*n^2 + 9*n)*x^2 + (n^2 + 18*n)*x^3 + n*x^4)); 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-ed9434bfd889c409","steps":["x \u21a6 x+1","D+1"],"edges":["e-0f7951445df45664","e-966d4ff49e200cc9"],"text":"A081579 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A081579 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}