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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":"9c407dd20340c9bae5af787e46d4b7a8b2f22705127316428eb55e2d25cc67f1","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-f562ab3e3b669a1b","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^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:         11,           60;\n  2:        121,         1320,          2025;\n  3:       1331,        21780,         66825,         54000;\n  4:      14641,       319440,       1470150,       2376000,        1265625;\n  5:     161051,      4392300,      26952750,      65340000,       69609375,       27337500;\n  6:    1771561,     57978360,     444720375,    1437480000,     2297109375,     1804275000,      558140625;\n  7:   19487171,    744055620,    6848693775,   27671490000,    58959140625,    69464587500,    42976828125,    10935000000;\n  8:  214358881,   9353842080,  100447508700,  487018224000,  1297101093750,  2037627900000,  1890980437500,   962280000000,   207594140625;\n  9: 2357947691, 115753795740, 1420614765900, 8035800696000, 25682601656250, 50431290525000, 62402354437500, 47632860000000, 20551819921875, 3844335937500;\n","recurrence_sentence":"The row polynomials satisfy (121+(495*n-165)*x+225*n^2*x^2)*P(n,x) = (1331+(5445*n+5445)*x+(2475*n^2+12375*n+7425)*x^2+(3375*n^2+6750*n+3375)*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[((1331 + (5445*n + 5445)*x + (2475*n^2 + 12375*n + 7425)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*P[n - 1, x])/(121 + (495*n - 165)*x + 225*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] = ((1331 + (5445*n + 5445)*x + (2475*n^2 + 12375*n + 7425)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*v[i - 1])/(121 + (495*n - 165)*x + 225*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-e6ebe74518d42cbe","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-aef71e51dba5316b","e-a615dab5832c157d"],"text":"A038259 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038259 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}