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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":"196ddfe0329f76c17c3e423a5b99acbae282dadfa8932bc1a7237b8e01e8e483","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) array.","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,  24,    9;\n  3: 1,  44,   99,    16;\n  4: 1,  64,  414,   256,     25;\n  5: 1,  84,  954,  1696,    525,     36;\n  6: 1, 104, 1719,  6336,   4775,    936,     49;\n  7: 1, 124, 2709, 16176,  25275,  10836,   1519,    64;\n  8: 1, 144, 3924, 33216,  90150,  74736,  21364,  2304,   81;\n  9: 1, 164, 5364, 59456, 243150, 350136, 182084, 38144, 3321, 100;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+n+(15*n^3-2*n^2-13*n)*x+(25*n^4-50*n^2-7*n)*x^2-(25*n^4+120*n^3+158*n^2+81*n)*x^3+(5*n^3+9*n^2)*x^4)*P(n,x) = -(n^2+n+(15*n^3+38*n^2-9*n-36)*x+(25*n^4+250*n^3+140*n^2-357*n-90)*x^2+(225*n^4+755*n^3+227*n^2-945*n-774)*x^3-(120*n^3+466*n^2+618*n+318)*x^4+(10*n^2+28*n+18)*x^5)*P(n-1,x) + ((n+1)*x+(15*n^2+47*n+27)*x^2+(25*n^3+385*n^2+687*n+242)*x^3+(450*n^3+2080*n^2+2967*n+1422)*x^4+(25*n^3+25*n^2-88*n-83)*x^5-(5*n^2+14*n+9)*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 + (15*n^3 + 38*n^2 - 9*n - 36)*x + (25*n^4 + 250*n^3 + 140*n^2 - 357*n - 90)*x^2 + (225*n^4 + 755*n^3 + 227*n^2 - 945*n - 774)*x^3 - (120*n^3 + 466*n^2 + 618*n + 318)*x^4 + (10*n^2 + 28*n + 18)*x^5)*P[n - 1, x] + ((n + 1)*x + (15*n^2 + 47*n + 27)*x^2 + (25*n^3 + 385*n^2 + 687*n + 242)*x^3 + (450*n^3 + 2080*n^2 + 2967*n + 1422)*x^4 + (25*n^3 + 25*n^2 - 88*n - 83)*x^5 - (5*n^2 + 14*n + 9)*x^6)*D[P[n - 1, x], x])/(-n^2 - n - (15*n^3 - 2*n^2 - 13*n)*x - (25*n^4 - 50*n^2 - 7*n)*x^2 + (25*n^4 + 120*n^3 + 158*n^2 + 81*n)*x^3 - (5*n^3 + 9*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 + (15*n^3 + 38*n^2 - 9*n - 36)*x + (25*n^4 + 250*n^3 + 140*n^2 - 357*n - 90)*x^2 + (225*n^4 + 755*n^3 + 227*n^2 - 945*n - 774)*x^3 - (120*n^3 + 466*n^2 + 618*n + 318)*x^4 + (10*n^2 + 28*n + 18)*x^5)*v[i - 1] + ((n + 1)*x + (15*n^2 + 47*n + 27)*x^2 + (25*n^3 + 385*n^2 + 687*n + 242)*x^3 + (450*n^3 + 2080*n^2 + 2967*n + 1422)*x^4 + (25*n^3 + 25*n^2 - 88*n - 83)*x^5 - (5*n^2 + 14*n + 9)*x^6)*deriv(v[i - 1], x))/(-n^2 - n - (15*n^3 - 2*n^2 - 13*n)*x - (25*n^4 - 50*n^2 - 7*n)*x^2 + (25*n^4 + 120*n^3 + 158*n^2 + 81*n)*x^3 - (5*n^3 + 9*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-ed9434bfd889c409","steps":["(\u03b8+1)\u00b2"],"edges":["e-83f3bc33323b8934"],"text":"A081579 \u2192 (\u03b8+1)\u00b2"},"headline":"A081579 \u2192 (\u03b8+1)\u00b2","chain":null}