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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":"bf72387b5ab05fbab110e82ed5a30893bacaf5ac41a3840012c4b43d40b07e19","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-8eb9bb3a09f0b7fd","role":"sequence","name":"Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (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,  12,   12;\n  1: 1,  24,   60,    32;\n  2: 1,  40,  180,   224,     80;\n  3: 1,  60,  420,   896,    720,     192;\n  4: 1,  84,  840,  2688,   3600,    2112,     448;\n  5: 1, 112, 1512,  6720,  13200,   12672,    5824,    1024;\n  6: 1, 144, 2520, 14784,  39600,   54912,   40768,   15360,    2304;\n  7: 1, 180, 3960, 29568, 102960,  192192,  203840,  122880,   39168,   5120;\n  8: 1, 220, 5940, 54912, 240240,  576576,  815360,  696320,  352512,  97280,  11264;\n  9: 1, 264, 8580, 96096, 514800, 1537536, 2772224, 3133440, 2232576, 972800, 236544, 24576;\n","recurrence_sentence":"The row polynomials satisfy -(6*n+9-(2*n^3+9*n^2+13*n+6)*x)*P(n,x) = -(6*n+9-(2*n^3+9*n^2+n-18)*x-(2*n^3+12*n^2+22*n+12)*x^2)*P(n-1,x) - (12*x-(4*n^2+16*n+6)*x^2-(2*n^2+8*n+6)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 12*x^2.","mathematica":"P[0, x_] := 1 + 12*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n + 9 - (2*n^3 + 9*n^2 + n - 18)*x - (2*n^3 + 12*n^2 + 22*n + 12)*x^2)*P[n - 1, x] - (12*x - (4*n^2 + 16*n + 6)*x^2 - (2*n^2 + 8*n + 6)*x^3)*D[P[n - 1, x], x])/(-6*n - 9 + (2*n^3 + 9*n^2 + 13*n + 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(6*n + 9 - (2*n^3 + 9*n^2 + n - 18)*x - (2*n^3 + 12*n^2 + 22*n + 12)*x^2)*v[i - 1] - (12*x - (4*n^2 + 16*n + 6)*x^2 - (2*n^2 + 8*n + 6)*x^3)*deriv(v[i - 1], x))/(-6*n - 9 + (2*n^3 + 9*n^2 + 13*n + 6)*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-8eb9bb3a09f0b7fd","steps":["\u03b8+1"],"edges":["e-5215afd4ce728407"],"text":"A123519 \u2192 \u03b8+1"},"headline":"A123519 \u2192 \u03b8+1","chain":null}