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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":"ae734a668a21ef1d29624aa80b8782c38062d1f701feb94634eea33040ce23f0","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-a313e9d783515c70","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+3*x)^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:      4,       18;\n  2:     16,      144,       243;\n  3:     64,      864,      2916,       2916;\n  4:    256,     4608,     23328,      46656,      32805;\n  5:   1024,    23040,    155520,     466560,     656100,      354294;\n  6:   4096,   110592,    933120,    3732480,    7873200,     8503056,     3720087;\n  7:  16384,   516096,   5225472,   26127360,   73483200,   119042784,   104162436,    38263752;\n  8:  65536,  2359296,  27869184,  167215104,  587865600,  1269789696,  1666598976,  1224440064,   387420489;\n  9: 262144, 10616832, 143327232, 1003290624, 4232632320, 11428107264, 19999187712, 22039921152, 13947137604, 3874204890;\n","recurrence_sentence":"The row polynomials satisfy (4+9*n*x)*P(n,x) = (16+(36*n+72)*x+(81*n+81)*x^2)*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[((16 + (36*n + 72)*x + (81*n + 81)*x^2)*P[n - 1, x])/(4 + 9*n*x)]];\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] = ((16 + (36*n + 72)*x + (81*n + 81)*x^2)*v[i - 1])/(4 + 9*n*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-6a5af5e1168bb679","steps":["basis A013610","\u03b8+1"],"edges":["e-39440d1ff583ab52","e-d0bbc4dbd2608b95"],"text":"A013610 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A013610 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}