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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":"c41934a85ae637b05cee0917ad4c04f244fff1905176412c5451849fcf64b342","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-c826b75746f15e6c","role":"sequence","name":"reversal of Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.","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,  2;\n  2: 1,  6,    6;\n  3: 1, 12,   30,    20;\n  4: 1, 20,   93,   148,     75;\n  5: 1, 30,  225,   640,    755,    312;\n  6: 1, 42,  465,  2080,   4280,   4044,    1421;\n  7: 1, 56,  861,  5600,  17780,  28812,   22841,    7016;\n  8: 1, 72, 1470, 13160,  59955, 147756,  198359,  136056,  37260;\n  9: 1, 90, 2358, 27888, 173355, 606186, 1223635, 1407168, 853452, 211470;\n","recurrence_sentence":"The row polynomials satisfy (n+(2*n-2)*x)*P(n,x) = -(n+1+(2*n^2+2*n-2)*x+(n^3+6*n^2-8*n-5)*x^2+(4*n^3-14*n)*x^3+(5*n^3-10*n^2-n+2)*x^4+(2*n^3-6*n^2+4*n)*x^5)*P(n-2,x) + ((n+1)*x^2+(n^2+4*n+1)*x^3+(4*n^2+3*n-1)*x^4+(5*n^2-2*n-1)*x^5+(2*n^2-2*n)*x^6)*d/dx P(n-2,x) + (2*n+1+(2*n^2+4*n-2)*x+(6*n^2-7*n+1)*x^2+(4*n^2-10*n+6)*x^3)*P(n-1,x) - (n*x^2+(3*n-2)*x^3+(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n - 2)*x + (n^3 + 6*n^2 - 8*n - 5)*x^2 + (4*n^3 - 14*n)*x^3 + (5*n^3 - 10*n^2 - n + 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + 4*n + 1)*x^3 + (4*n^2 + 3*n - 1)*x^4 + (5*n^2 - 2*n - 1)*x^5 + (2*n^2 - 2*n)*x^6)*D[P[n - 2, x], x] + (2*n + 1 + (2*n^2 + 4*n - 2)*x + (6*n^2 - 7*n + 1)*x^2 + (4*n^2 - 10*n + 6)*x^3)*P[n - 1, x] - (n*x^2 + (3*n - 2)*x^3 + (2*n - 2)*x^4)*D[P[n - 1, x], x])/(n + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^2 + 2*n - 2)*x + (n^3 + 6*n^2 - 8*n - 5)*x^2 + (4*n^3 - 14*n)*x^3 + (5*n^3 - 10*n^2 - n + 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*v[i - 2] + ((n + 1)*x^2 + (n^2 + 4*n + 1)*x^3 + (4*n^2 + 3*n - 1)*x^4 + (5*n^2 - 2*n - 1)*x^5 + (2*n^2 - 2*n)*x^6)*deriv(v[i - 2], x) + (2*n + 1 + (2*n^2 + 4*n - 2)*x + (6*n^2 - 7*n + 1)*x^2 + (4*n^2 - 10*n + 6)*x^3)*v[i - 1] - (n*x^2 + (3*n - 2)*x^3 + (2*n - 2)*x^4)*deriv(v[i - 1], x))/(n + (2*n - 2)*x)); 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-c15181c2ebe13c1d","steps":["reverse","\u03b8+1"],"edges":["e-130954a7e277f383","e-270cd506caddaedf"],"text":"A049020 \u2192 reverse \u2192 \u03b8+1"},"headline":"A049020 \u2192 reverse \u2192 \u03b8+1","chain":null}