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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":"1c0fd0af26768c2341556cfa1278241343f8494d5f4d972583ef253156fb0bc8","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-4c2a0f8cb422ddaa","role":"sequence","name":"Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 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,      16;\n  2: 1,      96,        144;\n  3: 1,     496,       2160,        1024;\n  4: 1,    2496,      23904,       28672,        6400;\n  5: 1,   12496,     237600,      542720,      288000,       36864;\n  6: 1,   62496,    2250864,     8744960,     8288000,     2433024,      200704;\n  7: 1,  312496,   20820240,   129690624,   195552000,    98832384,    18264064,    1048576;\n  8: 1, 1562496,  190194624,  1834033152,  4134950400,  3201859584,   994689024,  125829120,   5308416;\n  9: 1, 7812496, 1725814080, 25194926080, 81756864000, 91056365568, 42299572224, 8845787136, 812187648, 26214400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3-16*x+144*x^2)*P(n-2,x) - (8*x-16*x^2-64*x^3)*d/dx P(n-2,x) - (16*x^2-64*x^3)*d^2/dx^2 P(n-2,x) - (2-16*x)*P(n-1,x) + (8*x-16*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 16*x;\nP[n_, x_] := P[n, x] = Expand[(3 - 16*x + 144*x^2)*P[n - 2, x] - (8*x - 16*x^2 - 64*x^3)*D[P[n - 2, x], x] - (16*x^2 - 64*x^3)*D[P[n - 2, x], {x, 2}] - (2 - 16*x)*P[n - 1, x] + (8*x - 16*x^2)*D[P[n - 1, x], 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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (3 - 16*x + 144*x^2)*v[i - 2] - (8*x - 16*x^2 - 64*x^3)*deriv(v[i - 2], x) - (16*x^2 - 64*x^3)*deriv(deriv(v[i - 2], x), x) - (2 - 16*x)*v[i - 1] + (8*x - 16*x^2)*deriv(v[i - 1], 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-4c2a0f8cb422ddaa","steps":["(\u03b8+1)\u00b2"],"edges":["e-71b99e2f735f0c5e"],"text":"A285061 \u2192 (\u03b8+1)\u00b2"},"headline":"A285061 \u2192 (\u03b8+1)\u00b2","chain":null}