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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":"4040d5988943d4ca75fdce29997fc596843f99b7d923db522e1975d25c13f127","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-0088aa50f9dba3f3","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          13,           16;\n  2:         169,          416,          144;\n  3:        2197,         8112,         5616,         1024;\n  4:       28561,       140608,       146016,        53248,         6400;\n  5:      371293,      2284880,      3163680,      1730560,       416000,        36864;\n  6:     4826809,     35644128,     61691760,     44994560,     16224000,      2875392,      200704;\n  7:    62748517,    540602608,   1122790032,   1023626240,    492128000,    130830336,    18264064,    1048576;\n  8:   815730721,   8031810176,  19461693888,  21291425792,  12795328000,   4535451648,   949731328,  109051904,   5308416;\n  9: 10604499373, 117465223824, 325288312128, 415182802944, 299410675200, 132661960704, 37039521792, 6379536384, 621084672, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (169+(156*n-52)*x+16*n^2*x^2)*P(n,x) = (2197+(2028*n+2028)*x+(208*n^2+1040*n+624)*x^2+(64*n^2+128*n+64)*x^3)*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[((2197 + (2028*n + 2028)*x + (208*n^2 + 1040*n + 624)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(169 + (156*n - 52)*x + 16*n^2*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] = ((2197 + (2028*n + 2028)*x + (208*n^2 + 1040*n + 624)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(169 + (156*n - 52)*x + 16*n^2*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-81e6fb4f9f226d77","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-950a320de9db6fb9","e-90ab1e70686068cc"],"text":"A038327 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A038327 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}