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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":"d324a84b622d80f6c328fb866e64ab8f698a26c9545080915e1bd8c70f8d0e2b","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-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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:  2,   1;\n  2:  5,   6,    1;\n  3:  8,  21,   10,    1;\n  4: 11,  54,   52,   14,     1;\n  5: 14, 105,  184,   98,    18,    1;\n  6: 17, 174,  487,  452,   159,   22,    1;\n  7: 20, 261, 1042, 1535,   912,  235,   26,   1;\n  8: 23, 366, 1930, 4130,  3816, 1618,  326,  30,  1;\n  9: 26, 489, 3232, 9344, 12516, 8086, 2624, 432, 34, 1;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3-7*n^2+4*n-(3*n^3-10*n^2+22*n)*x-(8*n^2-19*n)*x^2-(n^2+6*n)*x^3-n*x^4)*P(n,x) = -(3*n^3-4*n^2+n+(15*n^3-44*n^2-n)*x+(37*n^2-99*n+69)*x^2+(5*n^2+23*n-31)*x^3+(4*n-13)*x^4-x^5)*P(n-1,x) + ((3*n^2-n)*x+(30*n^2-n-4)*x^2+(3*n^2+90*n-31)*x^3+(19*n+87)*x^4+(n+19)*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[(-(3*n^3 - 4*n^2 + n + (15*n^3 - 44*n^2 - n)*x + (37*n^2 - 99*n + 69)*x^2 + (5*n^2 + 23*n - 31)*x^3 + (4*n - 13)*x^4 - x^5)*P[n - 1, x] + ((3*n^2 - n)*x + (30*n^2 - n - 4)*x^2 + (3*n^2 + 90*n - 31)*x^3 + (19*n + 87)*x^4 + (n + 19)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 + 7*n^2 - 4*n + (3*n^3 - 10*n^2 + 22*n)*x + (8*n^2 - 19*n)*x^2 + (n^2 + 6*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] = (-(3*n^3 - 4*n^2 + n + (15*n^3 - 44*n^2 - n)*x + (37*n^2 - 99*n + 69)*x^2 + (5*n^2 + 23*n - 31)*x^3 + (4*n - 13)*x^4 - x^5)*v[i - 1] + ((3*n^2 - n)*x + (30*n^2 - n - 4)*x^2 + (3*n^2 + 90*n - 31)*x^3 + (19*n + 87)*x^4 + (n + 19)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 + 7*n^2 - 4*n + (3*n^3 - 10*n^2 + 22*n)*x + (8*n^2 - 19*n)*x^2 + (n^2 + 6*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-f25cd955a876cde3","steps":["D+1"],"edges":["e-303886c645549307"],"text":"A081577 \u2192 D+1"},"headline":"A081577 \u2192 D+1","chain":null}