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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":"7d3a8e34152264eaa9721060b0904bbeea8c52e788f06083f0e67890f4181f5b","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-d8f9ad9ce0e039d0","role":"sequence","name":"Pascal-(1,3,1) 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:  2,   1;\n  2:  6,   7,    1;\n  3: 10,  27,   12,     1;\n  4: 14,  79,   72,    17,     1;\n  5: 18, 163,  292,   141,    22,     1;\n  6: 22, 279,  864,   761,   234,    27,    1;\n  7: 26, 427, 1980,  2965,  1598,   351,   32,   1;\n  8: 30, 607, 3832,  8865,  7826,  2915,  492,  37,  1;\n  9: 34, 819, 6612, 21597, 29286, 17319, 4824, 657, 42, 1;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^3-10*n^2+6*n-(4*n^3-15*n^2+32*n)*x-(12*n^2-29*n)*x^2-(n^2+10*n)*x^3-n*x^4)*P(n,x) = -(4*n^3-6*n^2+2*n+(28*n^3-89*n^2+19*n)*x+(80*n^2-206*n+139)*x^2+(7*n^2+59*n-71)*x^3+(6*n-19)*x^4-x^5)*P(n-1,x) + ((4*n^2-2*n)*x+(56*n^2-15*n-6)*x^2+(4*n^2+181*n-71)*x^3+(27*n+177)*x^4+(n+27)*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[(-(4*n^3 - 6*n^2 + 2*n + (28*n^3 - 89*n^2 + 19*n)*x + (80*n^2 - 206*n + 139)*x^2 + (7*n^2 + 59*n - 71)*x^3 + (6*n - 19)*x^4 - x^5)*P[n - 1, x] + ((4*n^2 - 2*n)*x + (56*n^2 - 15*n - 6)*x^2 + (4*n^2 + 181*n - 71)*x^3 + (27*n + 177)*x^4 + (n + 27)*x^5 + x^6)*D[P[n - 1, x], x])/(-4*n^3 + 10*n^2 - 6*n + (4*n^3 - 15*n^2 + 32*n)*x + (12*n^2 - 29*n)*x^2 + (n^2 + 10*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] = (-(4*n^3 - 6*n^2 + 2*n + (28*n^3 - 89*n^2 + 19*n)*x + (80*n^2 - 206*n + 139)*x^2 + (7*n^2 + 59*n - 71)*x^3 + (6*n - 19)*x^4 - x^5)*v[i - 1] + ((4*n^2 - 2*n)*x + (56*n^2 - 15*n - 6)*x^2 + (4*n^2 + 181*n - 71)*x^3 + (27*n + 177)*x^4 + (n + 27)*x^5 + x^6)*deriv(v[i - 1], x))/(-4*n^3 + 10*n^2 - 6*n + (4*n^3 - 15*n^2 + 32*n)*x + (12*n^2 - 29*n)*x^2 + (n^2 + 10*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-d8f9ad9ce0e039d0","steps":["D+1"],"edges":["e-4c6b05990c1dd2cb"],"text":"A081578 \u2192 D+1"},"headline":"A081578 \u2192 D+1","chain":null}