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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":"89904315f1d4ba40978691f23135bbacee46085429739384a3929354cb7fa677","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-540fd96c9759fa70","role":"sequence","name":"A skewed version of triangular array A122075.","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: 0, 4;\n  2: 0, 2, 9;\n  3: 0, 0, 9, 20;\n  4: 0, 0, 3, 28, 40;\n  5: 0, 0, 0, 16, 75,  78;\n  6: 0, 0, 0,  4, 60, 180, 147;\n  7: 0, 0, 0,  0, 25, 186, 406,  272;\n  8: 0, 0, 0,  0,  5, 108, 511,  872,  495;\n  9: 0, 0, 0,  0,  0,  36, 378, 1296, 1809, 890;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (2+10*x+18*x^2+14*x^3)*P(n-3,x) + (x+5*x^2+9*x^3+5*x^4)*d/dx P(n-3,x) + (2+8*x+6*x^2)*P(n-2,x) + (x+4*x^2+5*x^3)*d/dx P(n-2,x) + (2+8*x)*P(n-1,x) - (1+4*x+5*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4*x, P(2,x) = 2*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[2, x_] := 2*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + 10*x + 18*x^2 + 14*x^3)*P[n - 3, x] + (x + 5*x^2 + 9*x^3 + 5*x^4)*D[P[n - 3, x], x] + (2 + 8*x + 6*x^2)*P[n - 2, x] + (x + 4*x^2 + 5*x^3)*D[P[n - 2, x], x] + (2 + 8*x)*P[n - 1, x] - (1 + 4*x + 5*x^2)*D[P[n - 1, x], x])/(2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; v[3] = 2*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2 + 10*x + 18*x^2 + 14*x^3)*v[i - 3] + (x + 5*x^2 + 9*x^3 + 5*x^4)*deriv(v[i - 3], x) + (2 + 8*x + 6*x^2)*v[i - 2] + (x + 4*x^2 + 5*x^3)*deriv(v[i - 2], x) + (2 + 8*x)*v[i - 1] - (1 + 4*x + 5*x^2)*deriv(v[i - 1], x))/(2)); 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-540fd96c9759fa70","steps":["\u03b8+1"],"edges":["e-93856e6fe1f734c5"],"text":"A236076 \u2192 \u03b8+1"},"headline":"A236076 \u2192 \u03b8+1","chain":null}