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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":"eb3bbf6e0a88b49309ab506a3114c1f492e6fc42b13e6e4e50f07012bb6749ef","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2e0409c0014ec240","role":"sequence","name":"Triangle of coefficients in expansion of (1+4x)^n.","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,  16;\n  2: 1,  32,  144;\n  3: 1,  48,  432,  1024;\n  4: 1,  64,  864,  4096,   6400;\n  5: 1,  80, 1440, 10240,  32000,   36864;\n  6: 1,  96, 2160, 20480,  96000,  221184,   200704;\n  7: 1, 112, 3024, 35840, 224000,  774144,  1404928,  1048576;\n  8: 1, 128, 4032, 57344, 448000, 2064384,  5619712,  8388608,  5308416;\n  9: 1, 144, 5184, 86016, 806400, 4644864, 16859136, 37748736, 47775744, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (1+(12*n-4)*x+16*n^2*x^2)*P(n,x) = (1+(12*n+12)*x+(16*n^2+80*n+48)*x^2+(64*n^2+128*n+64)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (12*n + 12)*x + (16*n^2 + 80*n + 48)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(1 + (12*n - 4)*x + 16*n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (12*n + 12)*x + (16*n^2 + 80*n + 48)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(1 + (12*n - 4)*x + 16*n^2*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-2e0409c0014ec240","steps":["(\u03b8+1)\u00b2"],"edges":["e-738497c1b246462b"],"text":"A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A013611 \u2192 (\u03b8+1)\u00b2","chain":null}