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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":"65ac1e0fef19646c6db4ed5a33d4d814fd686aec25d6eee8ccda79928e7a5c38","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-353fb5a029e4e669","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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: 1,   4;\n  2: 1,  10,    6;\n  3: 1,  18,   27,    8;\n  4: 1,  28,   75,   52,    10;\n  5: 1,  40,  165,  196,    85,    12;\n  6: 1,  54,  315,  560,   405,   126,   14;\n  7: 1,  70,  546, 1344,  1425,   726,  175,   16;\n  8: 1,  88,  882, 2856,  4125,  3036, 1183,  232,  18;\n  9: 1, 108, 1350, 5544, 10395, 10296, 5733, 1800, 297, 20;\n","recurrence_sentence":"The row polynomials satisfy (12*n+6-(2*n^3+3*n^2+37*n-42)*x+(4*n^3-12*n^2+8*n)*x^2)*P(n,x) = (12*n+6-(2*n^3+3*n^2+25*n-102)*x+(3*n^3-19*n^2-24*n+40)*x^2+(2*n^3-4*n^2-2*n+4)*x^3)*P(n-1,x) + (24*x-(4*n^2+8*n+82)*x^2+(7*n^2-10*n-38)*x^3+(2*n^2-2*n-4)*x^4)*d/dx 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[((12*n + 6 - (2*n^3 + 3*n^2 + 25*n - 102)*x + (3*n^3 - 19*n^2 - 24*n + 40)*x^2 + (2*n^3 - 4*n^2 - 2*n + 4)*x^3)*P[n - 1, x] + (24*x - (4*n^2 + 8*n + 82)*x^2 + (7*n^2 - 10*n - 38)*x^3 + (2*n^2 - 2*n - 4)*x^4)*D[P[n - 1, x], x])/(12*n + 6 - (2*n^3 + 3*n^2 + 37*n - 42)*x + (4*n^3 - 12*n^2 + 8*n)*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] = ((12*n + 6 - (2*n^3 + 3*n^2 + 25*n - 102)*x + (3*n^3 - 19*n^2 - 24*n + 40)*x^2 + (2*n^3 - 4*n^2 - 2*n + 4)*x^3)*v[i - 1] + (24*x - (4*n^2 + 8*n + 82)*x^2 + (7*n^2 - 10*n - 38)*x^3 + (2*n^2 - 2*n - 4)*x^4)*deriv(v[i - 1], x))/(12*n + 6 - (2*n^3 + 3*n^2 + 37*n - 42)*x + (4*n^3 - 12*n^2 + 8*n)*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-353fb5a029e4e669","steps":["\u03b8+1"],"edges":["e-9e187d16084c6bf9"],"text":"A207607 \u2192 \u03b8+1"},"headline":"A207607 \u2192 \u03b8+1","chain":null}