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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":"ccda44cc709991b9ac888f7ec360888ff95b742a75b20497cc57172cfc84b263","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-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","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, 12;\n   3: 0,  4, 81;\n   4: 0,  0, 54, 432;\n   5: 0,  0,  9, 432, 2025;\n   6: 0,  0,  0, 144, 2700,  8748;\n   7: 0,  0,  0,  16, 1350, 14580, 35721;\n   8: 0,  0,  0,   0,  300,  9720, 71442, 139968;\n   9: 0,  0,  0,   0,   25,  3240, 59535, 326592,  531441;\n  10: 0,  0,  0,   0,    0,   540, 26460, 326592, 1417176, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-3*n-(27*n^2-54*n+27)*x)*P(n,x) = -(2*n^3-2*n^2-4*n-(45*n^2+9*n+18)*x+162*n*x^2)*P(n-1,x) + ((4*n^2+4*n)*x+(9*n^2-27*n)*x^2-81*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 2*n^2 - 4*n - (45*n^2 + 9*n + 18)*x + 162*n*x^2)*P[n - 1, x] + ((4*n^2 + 4*n)*x + (9*n^2 - 27*n)*x^2 - 81*n*x^3)*D[P[n - 1, x], x])/(3*n^3 - 3*n - (27*n^2 - 54*n + 27)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 2*n^2 - 4*n - (45*n^2 + 9*n + 18)*x + 162*n*x^2)*v[i - 1] + ((4*n^2 + 4*n)*x + (9*n^2 - 27*n)*x^2 - 81*n*x^3)*deriv(v[i - 1], x))/(3*n^3 - 3*n - (27*n^2 - 54*n + 27)*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-5d39164c4108e043","steps":["(\u03b8+1)\u00b2"],"edges":["e-ab38a7348371e4e3"],"text":"A099097 \u2192 (\u03b8+1)\u00b2"},"headline":"A099097 \u2192 (\u03b8+1)\u00b2","chain":null}