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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":"48054c49dba9cb6d12c0e7785cc933128364c4293ce35132ef2d65ce83da3428","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-8217f7f3de935aeb","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:      3,        8;\n  2:     11,       60,        60;\n  3:     45,      372,       756,       448;\n  4:    197,     2184,      6720,      8064,       3360;\n  5:    903,    12560,     51840,     94080,      79200,      25344;\n  6:   4279,    71604,    370980,    902880,    1148400,     741312,     192192;\n  7:  20793,   406700,   2536380,   7749280,   13213200,   12876864,    6726720,    1464320;\n  8: 103049,  2306832,  16824192,  61885824,  132452320,  172972800,  136328192,   59744256,  11202048;\n  9: 518859, 13081752, 109242432, 470031744, 1211202720, 1985098752, 2095801344, 1384980480, 522422784, 85995520;\n","recurrence_sentence":"The row polynomials satisfy -(n+2-(n^3+3*n^2+2*n)*x^2)*P(n,x) = -(3*n+6+(2*n^2+10*n+12)*x-(3*n^3+9*n^2+6*n)*x^2-(4*n^3+16*n^2+20*n+8)*x^3)*P(n-1,x) - (1+3*x-(2*n^2+4*n)*x^2-(6*n^2+12*n+6)*x^3-(4*n^2+8*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[(-(3*n + 6 + (2*n^2 + 10*n + 12)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (4*n^3 + 16*n^2 + 20*n + 8)*x^3)*P[n - 1, x] - (1 + 3*x - (2*n^2 + 4*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (4*n^2 + 8*n + 4)*x^4)*D[P[n - 1, x], x])/(-n - 2 + (n^3 + 3*n^2 + 2*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] = (-(3*n + 6 + (2*n^2 + 10*n + 12)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (4*n^3 + 16*n^2 + 20*n + 8)*x^3)*v[i - 1] - (1 + 3*x - (2*n^2 + 4*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (4*n^2 + 8*n + 4)*x^4)*deriv(v[i - 1], x))/(-n - 2 + (n^3 + 3*n^2 + 2*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-66e2c3448245b0b7","steps":["basis A013609","\u03b8+1"],"edges":["e-4e2df72103232439","e-0099c851c31d5d2c"],"text":"A033282 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A033282 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}