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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":"a9cbd89e9bf2233631cf2ef3917d0ae8faaa00facb6e09bc4a960327597f1ee3","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-73781d44d680ea54","role":"sequence","name":"Triangle read by rows giving successive states of cellular automaton generated by \"Rule 20\" initiated with a single ON (black) cell.","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, 9;\n  2: 0, 0, 0, 0, 25;\n  3: 0, 0, 0, 0,  0, 36, 49;\n  4: 0, 0, 0, 0,  0,  0,  0, 0, 81;\n  5: 0, 0, 0, 0,  0,  0,  0, 0,  0, 100, 121;\n  6: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0, 169;\n  7: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0,   0, 196, 225;\n  8: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0,   0,   0,   0, 0, 289;\n  9: 0, 0, 0, 0,  0,  0,  0, 0,  0,   0,   0, 0,   0,   0,   0, 0,   0, 324, 361;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-12*n^3+13*n^2-6*n+1)*P(n,x) = -((8*n^5-28*n^4+32*n^3-12*n^2)*x+(16*n^5-36*n^4+16*n^3+11*n^2-4*n-3)*x^2)*P(n-1,x) + ((4*n^4-8*n^3+4*n^2)*x^2+(8*n^4-8*n^3-2*n^2+2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-((8*n^5 - 28*n^4 + 32*n^3 - 12*n^2)*x + (16*n^5 - 36*n^4 + 16*n^3 + 11*n^2 - 4*n - 3)*x^2)*P[n - 1, x] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (8*n^4 - 8*n^3 - 2*n^2 + 2*n + 1)*x^3)*D[P[n - 1, x], x])/(4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)]];\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 + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^5 - 28*n^4 + 32*n^3 - 12*n^2)*x + (16*n^5 - 36*n^4 + 16*n^3 + 11*n^2 - 4*n - 3)*x^2)*v[i - 1] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (8*n^4 - 8*n^3 - 2*n^2 + 2*n + 1)*x^3)*deriv(v[i - 1], x))/(4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)); 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-ddd13fb19530ec39","steps":["reverse (shift +1)","(\u03b8+1)\u00b2"],"edges":["e-e5594dfd8f34b1b6","e-f747f139cb76aa7d"],"text":"A133081 \u2192 reverse (shift +1) \u2192 (\u03b8+1)\u00b2"},"headline":"A133081 \u2192 reverse (shift +1) \u2192 (\u03b8+1)\u00b2","chain":null}