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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":"1fce4397a1a7abc61f9c3e2ed60036992674b30c19b363379dfe30f500cab03b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2cabfd060b04f15d","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-918795057d4e1130","kind":"oeis_match","node_a_id":"n-be708c90ebfa68eb","node_b_id":null,"fingerprint":null,"oeis_id":"A083410","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A083410","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A083410","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-be708c90ebfa68eb","oeis_id":"A083410","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":37897}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 4;\n   3: 0, 4,   18;\n   4: 0, 4,   54,     96;\n   5: 0, 4,  126,    576,     600;\n   6: 0, 4,  270,   2400,    6000,     4320;\n   7: 0, 4,  558,   8640,   39000,    64800,    35280;\n   8: 0, 4, 1134,  28896,  210000,   604800,   740880,    322560;\n   9: 0, 4, 2286,  92736, 1020600,  4536000,  9384480,   9031680,   3265920;\n  10: 0, 4, 4590, 290400, 4662000, 30028320, 93350880, 149022720, 117573120, 36288000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*x-2*x^2)*P(n-2,x) + (x-x^3)*d/dx P(n-2,x) - (1-4*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[(4*x - 2*x^2)*P[n - 2, x] + (x - x^3)*D[P[n - 2, x], x] - (1 - 4*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = (4*x - 2*x^2)*v[i - 2] + (x - x^3)*deriv(v[i - 2], x) - (1 - 4*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2cabfd060b04f15d","steps":["(\u03b8+1)\u00b2"],"edges":["e-60eb617d7ec68f2c"],"text":"A131689 \u2192 (\u03b8+1)\u00b2"},"headline":"A131689 \u2192 (\u03b8+1)\u00b2","chain":null}