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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":"6e46c012e9259e660b1a0a9eac772b87444d85e97aade7e028a777d1649f744f","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-eed4f744ce063213","role":"sequence","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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,  6,    6;\n  2: 1, 12,   39,    48,    20;\n  3: 1, 18,   99,   252,   330,    216,      56;\n  4: 1, 24,  186,   720,  1605,   2160,    1736,     768,     144;\n  5: 1, 30,  300,  1560,  4925,  10098,   13790,   12480,    7200,     2400,      352;\n  6: 1, 36,  441,  2880, 11775,  32508,   62923,   86688,   84780,    57600,    25872,     6912,      832;\n  7: 1, 42,  609,  4788, 24045,  83538,  209671,  389112,  539154,   556920,   423192,   229824,    84448,    18816,    1920;\n  8: 1, 48,  804,  7392, 44030, 184464,  569380, 1330752, 2391561,  3326880,  3578960,  2951424,  1831648,   827904,  257280,   49152,   4352;\n  9: 1, 54, 1026, 10800, 74430, 365148, 1338288, 3774816, 8346969, 14625630, 20403702, 22648896, 19883136, 13632192, 7145280, 2764800, 744192, 124416, 9728;\n","recurrence_sentence":"The row polynomials satisfy (1+3*n*x+(4*n-2)*x^2)*P(n,x) = (1+(3*n+6)*x+(13*n+13)*x^2+(18*n+12)*x^3+(8*n+4)*x^4)*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[((1 + (3*n + 6)*x + (13*n + 13)*x^2 + (18*n + 12)*x^3 + (8*n + 4)*x^4)*P[n - 1, x])/(1 + 3*n*x + (4*n - 2)*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] = ((1 + (3*n + 6)*x + (13*n + 13)*x^2 + (18*n + 12)*x^3 + (8*n + 4)*x^4)*v[i - 1])/(1 + 3*n*x + (4*n - 2)*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-eed4f744ce063213","steps":["\u03b8+1"],"edges":["e-a3d01cc804feb291"],"text":"A200536 \u2192 \u03b8+1"},"headline":"A200536 \u2192 \u03b8+1","chain":null}