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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":"4d0f6370117edf8e85a594c2aedfd258f4ea5d08f830c4fdd9ac12a25270f7d6","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-4438653f9e73a43c","role":"sequence","name":"reversal of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..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: 1,    4;\n  2: 1,   10,    15;\n  3: 1,   20,    69,     60;\n  4: 1,   38,   228,    436,     260;\n  5: 1,   72,   672,   2124,    2720,    1218;\n  6: 1,  138,  1893,   8816,   18205,   17256,     6139;\n  7: 1,  268,  5247,  33884,  100445,  151248,   112791,    33120;\n  8: 1,  526, 14514, 125324,  500855, 1065786,  1249948,   763960,  190323;\n  9: 1, 1040, 40362, 455228, 2359135, 6656838, 10918516, 10419608, 5374395, 1159750;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(2*n-2)*x)*P(n,x) = -((n^3-6*n^2+12*n-7)*x^2+(4*n^3-20*n^2+30*n-8)*x^3+(5*n^3-20*n^2+21*n-2)*x^4+(2*n^3-6*n^2+4*n)*x^5)*P(n-2,x) + ((n^2-3*n+2)*x^2+(5*n^2-12*n+5)*x^3+(9*n^2-17*n+4)*x^4+(7*n^2-10*n+1)*x^5+(2*n^2-2*n)*x^6)*d/dx P(n-2,x) + (n-2-(2*n^2-12*n+14)*x-(6*n^2-23*n+17)*x^2-(4*n^2-10*n+6)*x^3)*P(n-1,x) + ((n-2)*x+(4*n-6)*x^2+(5*n-6)*x^3+(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 12*n - 7)*x^2 + (4*n^3 - 20*n^2 + 30*n - 8)*x^3 + (5*n^3 - 20*n^2 + 21*n - 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*P[n - 2, x] + ((n^2 - 3*n + 2)*x^2 + (5*n^2 - 12*n + 5)*x^3 + (9*n^2 - 17*n + 4)*x^4 + (7*n^2 - 10*n + 1)*x^5 + (2*n^2 - 2*n)*x^6)*D[P[n - 2, x], x] + (n - 2 - (2*n^2 - 12*n + 14)*x - (6*n^2 - 23*n + 17)*x^2 - (4*n^2 - 10*n + 6)*x^3)*P[n - 1, x] + ((n - 2)*x + (4*n - 6)*x^2 + (5*n - 6)*x^3 + (2*n - 2)*x^4)*D[P[n - 1, x], x])/(n - 2 + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n^3 - 6*n^2 + 12*n - 7)*x^2 + (4*n^3 - 20*n^2 + 30*n - 8)*x^3 + (5*n^3 - 20*n^2 + 21*n - 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*v[i - 2] + ((n^2 - 3*n + 2)*x^2 + (5*n^2 - 12*n + 5)*x^3 + (9*n^2 - 17*n + 4)*x^4 + (7*n^2 - 10*n + 1)*x^5 + (2*n^2 - 2*n)*x^6)*deriv(v[i - 2], x) + (n - 2 - (2*n^2 - 12*n + 14)*x - (6*n^2 - 23*n + 17)*x^2 - (4*n^2 - 10*n + 6)*x^3)*v[i - 1] + ((n - 2)*x + (4*n - 6)*x^2 + (5*n - 6)*x^3 + (2*n - 2)*x^4)*deriv(v[i - 1], x))/(n - 2 + (2*n - 2)*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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","reverse","\u03b8+1"],"edges":["e-236d0fb15dc6eca0","e-0bbf9a609051b56f","e-cace39a2cf617802"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse \u2192 \u03b8+1"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse \u2192 \u03b8+1","chain":null}