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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":"492ca2df22d6817cfafefcaada16a38e6dc38317fc3c7a826e2d10ee8f1e21c7","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-351cbe3ca6567d98","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","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: 6, 12, 3;\n 2: 36, 144, 144, 48, 5;\n 3: 216, 1296, 2268, 1728, 630, 108, 7;\n 4: 1296, 10368, 25920, 31104, 20520, 7776, 1680, 192, 9;\n 5: 7776, 77760, 252720, 414720, 399600, 241056, 93240, 23040, 3510, 300, 11;\n 6: 46656, 559872, 2239488, 4665600, 5929200, 4945536, 2806272, 1099008, 296460, 54000, 6336, 432, 13;\n 7: 279936, 3919104, 18615744, 47029248, 74299680, 79361856, 59757264, 32431104, 12805128, 3674160, 756756, 108864, 10374, 588, 15;\n 8: 1679616, 26873856, 147806208, 438939648, 829543680, 1081672704, 1018096128, 710166528, 372571488, 147951360, 44440704, 10015488, 1664208, 197568, 15840, 768, 17;\n 9: 10077696, 181398528, 1133740800, 3869835264, 8515653120, 13121160192, 14841646848, 12684460032, 8338033728, 4259226240, 1698488352, 528519168, 127606752, 23623488, 3285360, 331776, 22950, 972, 19;\n","recurrence_sentence":"The row polynomials satisfy (6+6*n*x+(2*n-1)*x^2)*P(n,x) = (36+(36*n+72)*x+(48*n+48)*x^2+(18*n+12)*x^3+(2*n+1)*x^4)*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[((36 + (36*n + 72)*x + (48*n + 48)*x^2 + (18*n + 12)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(6 + 6*n*x + (2*n - 1)*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] = ((36 + (36*n + 72)*x + (48*n + 48)*x^2 + (18*n + 12)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(6 + 6*n*x + (2*n - 1)*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-d3c3090977105807","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-33933faab90bc4be","e-759c2d89a1198f6d"],"text":"A272867 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A272867 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}