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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":"f06a29465e5e03273e1f14c965f04897aca0c755112651baa8c61cf281bdc3e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cf23198864b5678a","role":"sequence","name":"Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 2;\n   3: 0, 2, 3;\n   4: 0, 0, 9,  4;\n   5: 0, 0, 3, 24,  5;\n   6: 0, 0, 0, 20, 50,   6;\n   7: 0, 0, 0,  4, 75,  90,   7;\n   8: 0, 0, 0,  0, 35, 210, 147,    8;\n   9: 0, 0, 0,  0,  5, 168, 490,  224,   9;\n  10: 0, 0, 0,  0,  0,  54, 588, 1008, 324, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n^3-n^2-2*n-(4*n^3-18*n^2+26*n-12)*x)*P(n,x) = (n^3-n^2-2*n-(7*n^3-11*n^2-4*n-4)*x+(12*n^3-38*n^2+24*n+8)*x^2)*P(n-1,x) - ((2*n^2+2*n)*x-(10*n^2-6*n-4)*x^2+(8*n^2-8*n-4)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - 2*n - (7*n^3 - 11*n^2 - 4*n - 4)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*P[n - 1, x] - ((2*n^2 + 2*n)*x - (10*n^2 - 6*n - 4)*x^2 + (8*n^2 - 8*n - 4)*x^3)*D[P[n - 1, x], x])/(-n^3 + n^2 + 2*n + (4*n^3 - 18*n^2 + 26*n - 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = ((n^3 - n^2 - 2*n - (7*n^3 - 11*n^2 - 4*n - 4)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*v[i - 1] - ((2*n^2 + 2*n)*x - (10*n^2 - 6*n - 4)*x^2 + (8*n^2 - 8*n - 4)*x^3)*deriv(v[i - 1], x))/(-n^3 + n^2 + 2*n + (4*n^3 - 18*n^2 + 26*n - 12)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf23198864b5678a","steps":["\u03b8+1"],"edges":["e-8af9663d803b33ba"],"text":"A098158 \u2192 \u03b8+1"},"headline":"A098158 \u2192 \u03b8+1","chain":null}