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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":"a5dae28bfd44481a92f840dc61f8a252dd457d614eb28edc8dd06ef4a86e8be2","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-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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,  16,    9;\n  3: 1,  28,   63,    16;\n  4: 1,  40,  198,   160,    25;\n  5: 1,  52,  414,   736,   325,    36;\n  6: 1,  64,  711,  2176,  1975,   576,    49;\n  7: 1,  76, 1089,  4912,  7675,  4356,   931,    64;\n  8: 1,  88, 1548,  9376, 22150, 21096,  8428,  1408,   81;\n  9: 1, 100, 2088, 16000, 52150, 75096, 49000, 14848, 2025, 100;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+n+(9*n^3-2*n^2-9*n)*x+(9*n^4-6*n^3-30*n^2-3*n)*x^2-(9*n^4+42*n^3+46*n^2+25*n)*x^3+(3*n^3+5*n^2)*x^4)*P(n,x) = -(n^2+n+(9*n^3+22*n^2-5*n-20)*x+(9*n^4+84*n^3+36*n^2-133*n-34)*x^2+(45*n^4+129*n^3-37*n^2-229*n-150)*x^3-(42*n^3+154*n^2+190*n+94)*x^4+(6*n^2+16*n+10)*x^5)*P(n-1,x) + ((n+1)*x+(9*n^2+27*n+15)*x^2+(9*n^3+129*n^2+221*n+74)*x^3+(90*n^3+396*n^2+509*n+230)*x^4+(9*n^3+9*n^2-30*n-27)*x^5-(3*n^2+8*n+5)*x^6)*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[(-(n^2 + n + (9*n^3 + 22*n^2 - 5*n - 20)*x + (9*n^4 + 84*n^3 + 36*n^2 - 133*n - 34)*x^2 + (45*n^4 + 129*n^3 - 37*n^2 - 229*n - 150)*x^3 - (42*n^3 + 154*n^2 + 190*n + 94)*x^4 + (6*n^2 + 16*n + 10)*x^5)*P[n - 1, x] + ((n + 1)*x + (9*n^2 + 27*n + 15)*x^2 + (9*n^3 + 129*n^2 + 221*n + 74)*x^3 + (90*n^3 + 396*n^2 + 509*n + 230)*x^4 + (9*n^3 + 9*n^2 - 30*n - 27)*x^5 - (3*n^2 + 8*n + 5)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (9*n^3 - 2*n^2 - 9*n)*x - (9*n^4 - 6*n^3 - 30*n^2 - 3*n)*x^2 + (9*n^4 + 42*n^3 + 46*n^2 + 25*n)*x^3 - (3*n^3 + 5*n^2)*x^4)]];\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] = (-(n^2 + n + (9*n^3 + 22*n^2 - 5*n - 20)*x + (9*n^4 + 84*n^3 + 36*n^2 - 133*n - 34)*x^2 + (45*n^4 + 129*n^3 - 37*n^2 - 229*n - 150)*x^3 - (42*n^3 + 154*n^2 + 190*n + 94)*x^4 + (6*n^2 + 16*n + 10)*x^5)*v[i - 1] + ((n + 1)*x + (9*n^2 + 27*n + 15)*x^2 + (9*n^3 + 129*n^2 + 221*n + 74)*x^3 + (90*n^3 + 396*n^2 + 509*n + 230)*x^4 + (9*n^3 + 9*n^2 - 30*n - 27)*x^5 - (3*n^2 + 8*n + 5)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (9*n^3 - 2*n^2 - 9*n)*x - (9*n^4 - 6*n^3 - 30*n^2 - 3*n)*x^2 + (9*n^4 + 42*n^3 + 46*n^2 + 25*n)*x^3 - (3*n^3 + 5*n^2)*x^4)); 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-f25cd955a876cde3","steps":["(\u03b8+1)\u00b2"],"edges":["e-be4074d4da1f384a"],"text":"A081577 \u2192 (\u03b8+1)\u00b2"},"headline":"A081577 \u2192 (\u03b8+1)\u00b2","chain":null}