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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":"ca0368fd21d2dc201e75e34972d4a737031379e7c2d229f87fbbfc98f8149dd5","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-53205d2a41e99499","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= 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,  32;\n  2: 1,  64,  243;\n  3: 1,  96,  729,  1024;\n  4: 1, 128, 1458,  4096,   3125;\n  5: 1, 160, 2430, 10240,  15625,   7776;\n  6: 1, 192, 3645, 20480,  46875,  46656,   16807;\n  7: 1, 224, 5103, 35840, 109375, 163296,  117649,   32768;\n  8: 1, 256, 6804, 57344, 218750, 435456,  470596,  262144,  59049;\n  9: 1, 288, 8748, 86016, 393750, 979776, 1411788, 1179648, 531441, 100000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+6*x+15*x^2+20*x^3+15*x^4+6*x^5+x^6)*P(n-6,x) + (6+30*x+60*x^2+60*x^3+30*x^4+6*x^5)*P(n-5,x) - (15+60*x+90*x^2+60*x^3+15*x^4)*P(n-4,x) + (20+60*x+60*x^2+20*x^3)*P(n-3,x) - (15+30*x+15*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1 + 32*x, P(2,x) = 1 + 64*x + 243*x^2, P(3,x) = 1 + 96*x + 729*x^2 + 1024*x^3, P(4,x) = 1 + 128*x + 1458*x^2 + 4096*x^3 + 3125*x^4, P(5,x) = 1 + 160*x + 2430*x^2 + 10240*x^3 + 15625*x^4 + 7776*x^5.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 32*x;\nP[2, x_] := 1 + 64*x + 243*x^2;\nP[3, x_] := 1 + 96*x + 729*x^2 + 1024*x^3;\nP[4, x_] := 1 + 128*x + 1458*x^2 + 4096*x^3 + 3125*x^4;\nP[5, x_] := 1 + 160*x + 2430*x^2 + 10240*x^3 + 15625*x^4 + 7776*x^5;\nP[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6)*P[n - 6, x] + (6 + 30*x + 60*x^2 + 60*x^3 + 30*x^4 + 6*x^5)*P[n - 5, x] - (15 + 60*x + 90*x^2 + 60*x^3 + 15*x^4)*P[n - 4, x] + (20 + 60*x + 60*x^2 + 20*x^3)*P[n - 3, x] - (15 + 30*x + 15*x^2)*P[n - 2, x] + (6 + 6*x)*P[n - 1, 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 + 32*x; v[3] = 1 + 64*x + 243*x^2; v[4] = 1 + 96*x + 729*x^2 + 1024*x^3; v[5] = 1 + 128*x + 1458*x^2 + 4096*x^3 + 3125*x^4; v[6] = 1 + 160*x + 2430*x^2 + 10240*x^3 + 15625*x^4 + 7776*x^5; for(n = 6, N, my(i = n + 1); v[i] = -(1 + 6*x + 15*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6)*v[i - 6] + (6 + 30*x + 60*x^2 + 60*x^3 + 30*x^4 + 6*x^5)*v[i - 5] - (15 + 60*x + 90*x^2 + 60*x^3 + 15*x^4)*v[i - 4] + (20 + 60*x + 60*x^2 + 20*x^3)*v[i - 3] - (15 + 30*x + 15*x^2)*v[i - 2] + (6 + 6*x)*v[i - 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-e7d0ffa461d9dbc4","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-14eb908136aaeb50","e-8c57ee3d2c76f884"],"text":"A125092 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A125092 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}