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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":"896ac881674cce769a6381a09374e5a2c87b861228adc6aa2edf6e2fb6f1c849","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-7ce10652a7d44cf0","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:         4;\n  2:        34,         48;\n  3:       266,        784,         576;\n  4:      2196,      10024,       15216,        7680;\n  5:     19692,     122872,      286944,      297216,       115200;\n  6:    192816,    1534352,     4875864,     7734016,      6122880,      1935360;\n  7:   2057904,   19979280,    80707224,   173624576,    209786240,    134977536,     36126720;\n  8:  23844960,  273877344,  1346512800,  3673219136,   6004415360,   5881158144,   3195795456,   743178240;\n  9: 298582560, 3966449184, 23027961120, 76312418496, 157879047360, 208798708224, 172380773376, 81221713920, 16721510400;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(4*n^3+8*n^2+4*n)*x+(4*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(4*n^2+4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x + (4*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x + (4*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A013609","D"],"edges":["e-5f6e941a4a66ef73","e-519881674cae5a7f"],"text":"A145324 \u2192 basis A013609 \u2192 D"},"headline":"A145324 \u2192 basis A013609 \u2192 D","chain":null}