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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":"da0c9e90b88b1724556440cfd6d3b7d2bfe246bcdc94240a98a40f3b386bba52","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-56975226496d4c31","role":"sequence","name":"gamma_polynomial of Pascal-(1,6,1) 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:  1;\n  2:  7,    6;\n  3: 13,   12;\n  4: 19,   90,    36;\n  5: 25,  240,   108;\n  6: 31,  462,   864,   216;\n  7: 37,  756,  2952,   864;\n  8: 43, 1122,  7020,  7344, 1296;\n  9: 49, 1560, 13716, 30240, 6480;\n","recurrence_sentence":"The row polynomials satisfy (6*n^3-17*n^2+11*n+(24*n^2-59*n)*x+24*n*x^2)*P(n,x) = (6*n^3-11*n^2+5*n+(72*n^3-252*n^2+145*n)*x+(288*n^2-684*n+420)*x^2+(288*n-288)*x^3)*P(n-1,x) - ((6*n^2-5*n)*x+(144*n^2-96*n-11)*x^2+(576*n-240)*x^3+576*x^4)*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[((6*n^3 - 11*n^2 + 5*n + (72*n^3 - 252*n^2 + 145*n)*x + (288*n^2 - 684*n + 420)*x^2 + (288*n - 288)*x^3)*P[n - 1, x] - ((6*n^2 - 5*n)*x + (144*n^2 - 96*n - 11)*x^2 + (576*n - 240)*x^3 + 576*x^4)*D[P[n - 1, x], x])/(6*n^3 - 17*n^2 + 11*n + (24*n^2 - 59*n)*x + 24*n*x^2)]];\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] = ((6*n^3 - 11*n^2 + 5*n + (72*n^3 - 252*n^2 + 145*n)*x + (288*n^2 - 684*n + 420)*x^2 + (288*n - 288)*x^3)*v[i - 1] - ((6*n^2 - 5*n)*x + (144*n^2 - 96*n - 11)*x^2 + (576*n - 240)*x^3 + 576*x^4)*deriv(v[i - 1], x))/(6*n^3 - 17*n^2 + 11*n + (24*n^2 - 59*n)*x + 24*n*x^2)); 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-212fa067c32397af","steps":["\u03b3","D+1"],"edges":["e-7dd3cff0cd2ec979","e-b6e51e7083daaea7"],"text":"A081581 \u2192 \u03b3 \u2192 D+1"},"headline":"A081581 \u2192 \u03b3 \u2192 D+1","chain":null}