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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":"f90aa9836c645bac8bf02897f0f37a849e920b4b7097365357df7fa54c5dacb5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a34e8f08502aec80","role":"sequence","name":"derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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:       0;\n  1:       1;\n  2:       3,       4;\n  3:      11,      24,       9;\n  4:      50,     140,      90,      16;\n  5:     274,     900,     765,     240,      25;\n  6:    1764,    6496,    6615,    2800,     525,     36;\n  7:   13068,   52528,   60921,   31360,    8050,   1008,    49;\n  8:  109584,  472496,  605556,  359184,  113400,  19656,  1764,   64;\n  9: 1026576, 4690800, 6513120, 4309200, 1581825, 340200, 42630, 2880, 81;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (3*n-3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3 + 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (3*n - 3 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 3 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (3*n - 3 + 3*x)*v[i - 1]); 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-efe23a76953cf69d","steps":["D","\u03b8+1"],"edges":["e-04f1d27fb1fcfee8","e-9059b5c63c4bd219"],"text":"A130534 \u2192 D \u2192 \u03b8+1"},"headline":"A130534 \u2192 D \u2192 \u03b8+1","chain":null}