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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":"58b1027a05909dce50c072d5369f1953b74a853c35533b5a08595d8a89718cf7","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-7196be931c2ed996","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:  4,   3;\n  2:  5,  10,    3;\n  3:  8,  37,   42,    9;\n  4:  9,  52,   94,   60,     9;\n  5: 12, 103,  316,  414,   216,    27;\n  6: 13, 126,  465,  820,   711,   270,    27;\n  7: 16, 201, 1014, 2623,  3684,  2727,   918,    81;\n  8: 17, 232, 1308, 3944,  6886,  7032,  3996,  1080,   81;\n  9: 20, 331, 2328, 9068, 21376, 31290, 28008, 14364, 3564, 243;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2-11*n-11+(18*n^2+18*n-58)*x+(54*n-36)*x^2+36*x^3)*P(n,x) = -(16*n^3-108*n^2+106*n+26+(48*n^3-498*n^2+550*n+18)*x+(36*n^3-867*n^2+931*n-106)*x^2-(612*n^2-132*n-282)*x^3-(162*n^2+522*n-576)*x^4-(216*n-216)*x^5)*P(n-1,x) + (16*n^2-56*n-15+(112*n^2-456*n-122)*x+(276*n^2-1324*n-543)*x^2+(288*n^2-1608*n-1546)*x^3+(108*n^2-684*n-2262)*x^4-1368*x^5-216*x^6)*d/dx P(n-1,x) - (4*n+1+(44*n+12)*x+(184*n+72)*x^2+(360*n+256)*x^3+(324*n+501)*x^4+(108*n+468)*x^5+162*x^6)*d^2/dx^2 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[(-(16*n^3 - 108*n^2 + 106*n + 26 + (48*n^3 - 498*n^2 + 550*n + 18)*x + (36*n^3 - 867*n^2 + 931*n - 106)*x^2 - (612*n^2 - 132*n - 282)*x^3 - (162*n^2 + 522*n - 576)*x^4 - (216*n - 216)*x^5)*P[n - 1, x] + (16*n^2 - 56*n - 15 + (112*n^2 - 456*n - 122)*x + (276*n^2 - 1324*n - 543)*x^2 + (288*n^2 - 1608*n - 1546)*x^3 + (108*n^2 - 684*n - 2262)*x^4 - 1368*x^5 - 216*x^6)*D[P[n - 1, x], x] - (4*n + 1 + (44*n + 12)*x + (184*n + 72)*x^2 + (360*n + 256)*x^3 + (324*n + 501)*x^4 + (108*n + 468)*x^5 + 162*x^6)*D[P[n - 1, x], {x, 2}])/(12*n^2 - 11*n - 11 + (18*n^2 + 18*n - 58)*x + (54*n - 36)*x^2 + 36*x^3)]];\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] = (-(16*n^3 - 108*n^2 + 106*n + 26 + (48*n^3 - 498*n^2 + 550*n + 18)*x + (36*n^3 - 867*n^2 + 931*n - 106)*x^2 - (612*n^2 - 132*n - 282)*x^3 - (162*n^2 + 522*n - 576)*x^4 - (216*n - 216)*x^5)*v[i - 1] + (16*n^2 - 56*n - 15 + (112*n^2 - 456*n - 122)*x + (276*n^2 - 1324*n - 543)*x^2 + (288*n^2 - 1608*n - 1546)*x^3 + (108*n^2 - 684*n - 2262)*x^4 - 1368*x^5 - 216*x^6)*deriv(v[i - 1], x) - (4*n + 1 + (44*n + 12)*x + (184*n + 72)*x^2 + (360*n + 256)*x^3 + (324*n + 501)*x^4 + (108*n + 468)*x^5 + 162*x^6)*deriv(deriv(v[i - 1], x), x))/(12*n^2 - 11*n - 11 + (18*n^2 + 18*n - 58)*x + (54*n - 36)*x^2 + 36*x^3)); 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-7196be931c2ed996","steps":["D+1"],"edges":["e-c18e1ce17600bec1"],"text":"A152842 \u2192 D+1"},"headline":"A152842 \u2192 D+1","chain":null}