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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":"8952a43a77fb0bb93c1dbc9c8831d690e83e50315df75d295300efd8eaee84cf","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-de2147e0b2be5082","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see 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:     0;\n  1:     6;\n  2:    27,     36;\n  3:    99,    270,     162;\n  4:   333,   1368,    1701,      648;\n  5:  1065,   5832,   11097,     8748,     2430;\n  6:  3294,  22518,   57834,    69984,    40095,     8748;\n  7:  9948,  81486,  263736,   432540,   380295,   170586,   30618;\n  8: 29511, 281628, 1100061,  2282580,  2722815,  1872072,  688905,  104976;\n  9: 86337, 940464, 4303773, 10815444, 16347825, 15247764, 8588349, 2676888, 354294;\n","recurrence_sentence":"The row polynomials satisfy (6*n^3-45*n^2+45*n-6+(36*n^3-108*n^2+108*n-36)*x)*P(n,x) = (9*n^3-63*n^2-18*n+(63*n^3-144*n^2-27*n)*x+(54*n^3-54*n^2)*x^2)*P(n-1,x) + (5*n^2-20*n-15+(48*n^2-102*n-54)*x+(117*n^2-144*n-27)*x^2+(54*n^2-54*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 63*n^2 - 18*n + (63*n^3 - 144*n^2 - 27*n)*x + (54*n^3 - 54*n^2)*x^2)*P[n - 1, x] + (5*n^2 - 20*n - 15 + (48*n^2 - 102*n - 54)*x + (117*n^2 - 144*n - 27)*x^2 + (54*n^2 - 54*n)*x^3)*D[P[n - 1, x], x])/(6*n^3 - 45*n^2 + 45*n - 6 + (36*n^3 - 108*n^2 + 108*n - 36)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 63*n^2 - 18*n + (63*n^3 - 144*n^2 - 27*n)*x + (54*n^3 - 54*n^2)*x^2)*v[i - 1] + (5*n^2 - 20*n - 15 + (48*n^2 - 102*n - 54)*x + (117*n^2 - 144*n - 27)*x^2 + (54*n^2 - 54*n)*x^3)*deriv(v[i - 1], x))/(6*n^3 - 45*n^2 + 45*n - 6 + (36*n^3 - 108*n^2 + 108*n - 36)*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-353fb5a029e4e669","steps":["basis A013610","D"],"edges":["e-4d45de18adc3c05d","e-6f88cbcca0ef0df1"],"text":"A207607 \u2192 basis A013610 \u2192 D"},"headline":"A207607 \u2192 basis A013610 \u2192 D","chain":null}