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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":"e82c1b3d3ffc63f27cefc04c87e1aaceb1c2e671b31a365795544b0f95f23cdf","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-dd70177049267f47","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:     11,       4;\n  1:     44,      34,       6;\n  2:    155,     186,      69,       8;\n  3:    510,     834,     480,     116,      10;\n  4:   1608,    3336,    2622,     980,     175,     12;\n  5:   4924,   12390,   12390,    6320,    1740,    246,     14;\n  6:  14761,   43682,   53133,   34500,   12945,   2814,    329,    16;\n  7:  43540,  148178,  212532,  168024,   80220,  23730,   4256,   424,   18;\n  8: 126805,  487932,  806730,  752948,  437745, 164640,  40124,  6120,  531,  20;\n  9: 365532, 1569228, 2939556, 3166460, 2174200, 995484, 308196, 63792, 8460, 650, 22;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-4*n-6+(4*n^2+10*n+6)*x)*P(n,x) = (3*n^2-2*n-33+(7*n^2+28*n+21)*x+(2*n^2+10*n+12)*x^2)*P(n-1,x) + (5*n-5+(16*n+14)*x+(13*n+23)*x^2+(2*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 4*x.","mathematica":"P[0, x_] := 11 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*n - 33 + (7*n^2 + 28*n + 21)*x + (2*n^2 + 10*n + 12)*x^2)*P[n - 1, x] + (5*n - 5 + (16*n + 14)*x + (13*n + 23)*x^2 + (2*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 4*n - 6 + (4*n^2 + 10*n + 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 2*n - 33 + (7*n^2 + 28*n + 21)*x + (2*n^2 + 10*n + 12)*x^2)*v[i - 1] + (5*n - 5 + (16*n + 14)*x + (13*n + 23)*x^2 + (2*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 4*n - 6 + (4*n^2 + 10*n + 6)*x)); 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-dfab35176bd08e72","steps":["x \u21a6 x+1","D"],"edges":["e-23e971f1ed1ce0e7","e-1cec082a51821577"],"text":"A207606 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A207606 \u2192 x \u21a6 x+1 \u2192 D","chain":null}