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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":"19fbf5002642a75bf23e8118da68c8671b09a0623a432973f7beee225ae2cf1f","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-2fe6a26f73f75da6","role":"sequence","name":"basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    9,    24,     12;\n  1:   18,    84,     96,      32;\n  2:   36,   240,    444,     320,      80;\n  3:   72,   624,   1608,    1824,     960,     192;\n  4:  144,  1536,   5088,    7936,    6480,    2688,     448;\n  5:  288,  3648,  14784,   29440,   32800,   20928,    7168,    1024;\n  6:  576,  8448,  40512,   98304,  139200,  120576,   63168,   18432,    2304;\n  7: 1152, 19200, 106368,  304640,  524160,  575232,  407680,  181248,   46080,    5120;\n  8: 2304, 43008, 270336,  892928, 1809920, 2408448, 2157568, 1294336,  499968,  112640,  11264;\n  9: 4608, 95232, 669696, 2506752, 5852160, 9160704, 9934848, 7520256, 3912192, 1336320, 270336, 24576;\n","recurrence_sentence":"The row polynomials satisfy (3+(3*n+6)*x+(2*n+4)*x^2)*P(n,x) = (6+(6*n+24)*x+(10*n+30)*x^2+(4*n+12)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 12*x^2.","mathematica":"P[0, x_] := 9 + 24*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6 + (6*n + 24)*x + (10*n + 30)*x^2 + (4*n + 12)*x^3)*P[n - 1, x])/(3 + (3*n + 6)*x + (2*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 24*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + (6*n + 24)*x + (10*n + 30)*x^2 + (4*n + 12)*x^3)*v[i - 1])/(3 + (3*n + 6)*x + (2*n + 4)*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-f48bed27dbc29c5f","steps":["basis A013609","\u03b8+1"],"edges":["e-8866dbf598c953fc","e-efaa5d33ec0900d9"],"text":"A391533 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A391533 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}