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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"4cbb55f5b400d17b4c1ffc7eae46f6ec738d698e26a63bf4bedc9b44b29880f8","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-f24dc5f77f658331","role":"sequence","name":"Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","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:  2,   4;\n  2:  3,  16,    9;\n  3:  4,  36,   54,   16;\n  4:  5,  64,  162,  128,    25;\n  5:  6, 100,  360,  480,   250,    36;\n  6:  7, 144,  675, 1280,  1125,   432,    49;\n  7:  8, 196, 1134, 2800,  3500,  2268,   686,   64;\n  8:  9, 256, 1764, 5376,  8750,  8064,  4116, 1024,   81;\n  9: 10, 324, 2592, 9408, 18900, 22680, 16464, 6912, 1458, 100;\n","recurrence_sentence":"The row polynomials satisfy (n+(3*n^2-4*n+4)*x+(n^3-2*n^2+6*n-2)*x^2+n^2*x^3)*P(n,x) = (n+1+(3*n^2+3*n+4)*x+(n^3+4*n^2+7*n+6)*x^2+(n^3+2*n^2+7*n+4)*x^3+(n^2+2*n+1)*x^4)*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[((n + 1 + (3*n^2 + 3*n + 4)*x + (n^3 + 4*n^2 + 7*n + 6)*x^2 + (n^3 + 2*n^2 + 7*n + 4)*x^3 + (n^2 + 2*n + 1)*x^4)*P[n - 1, x])/(n + (3*n^2 - 4*n + 4)*x + (n^3 - 2*n^2 + 6*n - 2)*x^2 + n^2*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] = ((n + 1 + (3*n^2 + 3*n + 4)*x + (n^3 + 4*n^2 + 7*n + 6)*x^2 + (n^3 + 2*n^2 + 7*n + 4)*x^3 + (n^2 + 2*n + 1)*x^4)*v[i - 1])/(n + (3*n^2 - 4*n + 4)*x + (n^3 - 2*n^2 + 6*n - 2)*x^2 + n^2*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-c8c19eba9d68881f","steps":["D+1","(\u03b8+1)\u00b2"],"edges":["e-5c65a1a70531c253","e-a8b2755a293a1f99"],"text":"A007318 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}