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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":"458c395ad9cf2f4627c8a98ecfbe1140106c7c25f3d6be9d168b42aa35088a34","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-8348c590b954589f","role":"sequence","name":"one_plus_derivative 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:  5,   16;\n  2:  9,  104,    81;\n  3: 13,  264,   675,    256;\n  4: 17,  496,  2214,   2624,    625;\n  5: 21,  800,  5130,  10560,   7625,    1296;\n  6: 25, 1176,  9855,  29120,  36375,   18360,   2401;\n  7: 29, 1624, 16821,  64960, 116375,  101304,  38759,   4096;\n  8: 33, 2144, 26460, 126336, 295750,  368928, 242844,  74240,   6561;\n  9: 37, 2736, 39204, 223104, 645750, 1052352, 991956, 520704, 131949, 10000;\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) = 5 + 16*x, P(2,x) = 9 + 104*x + 81*x^2, P(3,x) = 13 + 264*x + 675*x^2 + 256*x^3, P(4,x) = 17 + 496*x + 2214*x^2 + 2624*x^3 + 625*x^4, P(5,x) = 21 + 800*x + 5130*x^2 + 10560*x^3 + 7625*x^4 + 1296*x^5.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 16*x;\nP[2, x_] := 9 + 104*x + 81*x^2;\nP[3, x_] := 13 + 264*x + 675*x^2 + 256*x^3;\nP[4, x_] := 17 + 496*x + 2214*x^2 + 2624*x^3 + 625*x^4;\nP[5, x_] := 21 + 800*x + 5130*x^2 + 10560*x^3 + 7625*x^4 + 1296*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] = 5 + 16*x; v[3] = 9 + 104*x + 81*x^2; v[4] = 13 + 264*x + 675*x^2 + 256*x^3; v[5] = 17 + 496*x + 2214*x^2 + 2624*x^3 + 625*x^4; v[6] = 21 + 800*x + 5130*x^2 + 10560*x^3 + 7625*x^4 + 1296*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":["D+1","(\u03b8+1)\u00b2"],"edges":["e-720363525db6d149","e-42bd0c31f484c042"],"text":"A125092 \u2192 D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A125092 \u2192 D+1 \u2192 (\u03b8+1)\u00b2","chain":null}