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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":"0a0bfa5cbc429972b4b8ef13bb9df1ac75ad613da196c60e082e0082fb9d836b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c9dce8c25621fdea","role":"sequence","name":"Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,    1;\n   3:  4,    5,     1;\n   4:  7,   22,    11,      1;\n   5: 11,   68,   101,     28,      1;\n   6: 16,  165,   510,    501,     94,       1;\n   7: 22,  341,  1795,   3801,   2889,     421,       1;\n   8: 29,  630,  5026,  18291,  30394,   19475,    2379,       1;\n   9: 37, 1072, 12026,  66472, 189959,  266568,  151122,   16080,      1;\n  10: 46, 1713, 25656, 199374, 863730, 2076747, 2570756, 1327074, 125682, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = -((6*n^2-24*n+18)*x^2+(2*n^3-22*n^2+74*n-78)*x^3+(n^3-10*n^2+31*n-30)*x^4)*P(n-3,x) - ((2*n^2-8*n+6)*x^3+(n^3-10*n^2+31*n-30)*x^4)*d/dx P(n-3,x) - (2*n-2-(6*n-10)*x+(3*n^2-11*n+10)*x^2-(n^3-9*n^2+24*n-20)*x^3)*P(n-2,x) + ((2*n-2)*x+(3*n^2-13*n+14)*x^2+(n^3-9*n^2+24*n-20)*x^3)*d/dx P(n-2,x) + (4*n-6+(4*n^2-18*n+18)*x-(n^2-7*n+10)*x^2)*P(n-1,x) - ((2*n-2)*x+(n^2-7*n+10)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 4 + 5*x + x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 4 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-((6*n^2 - 24*n + 18)*x^2 + (2*n^3 - 22*n^2 + 74*n - 78)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*P[n - 3, x] - ((2*n^2 - 8*n + 6)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*D[P[n - 3, x], x] - (2*n - 2 - (6*n - 10)*x + (3*n^2 - 11*n + 10)*x^2 - (n^3 - 9*n^2 + 24*n - 20)*x^3)*P[n - 2, x] + ((2*n - 2)*x + (3*n^2 - 13*n + 14)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*D[P[n - 2, x], x] + (4*n - 6 + (4*n^2 - 18*n + 18)*x - (n^2 - 7*n + 10)*x^2)*P[n - 1, x] - ((2*n - 2)*x + (n^2 - 7*n + 10)*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; v[3] = 4 + 5*x + x^2; for(n = 4, N, my(i = n); v[i] = (-((6*n^2 - 24*n + 18)*x^2 + (2*n^3 - 22*n^2 + 74*n - 78)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*v[i - 3] - ((2*n^2 - 8*n + 6)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*deriv(v[i - 3], x) - (2*n - 2 - (6*n - 10)*x + (3*n^2 - 11*n + 10)*x^2 - (n^3 - 9*n^2 + 24*n - 20)*x^3)*v[i - 2] + ((2*n - 2)*x + (3*n^2 - 13*n + 14)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*deriv(v[i - 2], x) + (4*n - 6 + (4*n^2 - 18*n + 18)*x - (n^2 - 7*n + 10)*x^2)*v[i - 1] - ((2*n - 2)*x + (n^2 - 7*n + 10)*x^2)*deriv(v[i - 1], x))/(2*n - 4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["D+1"],"edges":["e-778c0ce9b6b95629"],"text":"A046716 \u2192 D+1"},"headline":"A046716 \u2192 D+1","chain":null}