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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":"560a96a758798d0f57243dc7d3e90d3d3443f45529ed18762a93177853160c11","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-a920b56c9b3ef4d1","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","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:         10,          4;\n  2:        100,         80,          9;\n  3:       1000,       1200,        270,         16;\n  4:      10000,      16000,       5400,        640,        25;\n  5:     100000,     200000,      90000,      16000,      1250,       36;\n  6:    1000000,    2400000,    1350000,     320000,     37500,     2160,      49;\n  7:   10000000,   28000000,   18900000,    5600000,    875000,    75600,    3430,     64;\n  8:  100000000,  320000000,  252000000,   89600000,  17500000,  2016000,  137200,   5120,   81;\n  9: 1000000000, 3600000000, 3240000000, 1344000000, 315000000, 45360000, 4116000, 230400, 7290, 100;\n","recurrence_sentence":"The row polynomials satisfy (100+(30*n-10)*x+n^2*x^2)*P(n,x) = (1000+(300*n+300)*x+(10*n^2+50*n+30)*x^2+(n^2+2*n+1)*x^3)*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[((1000 + (300*n + 300)*x + (10*n^2 + 50*n + 30)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(100 + (30*n - 10)*x + 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] = ((1000 + (300*n + 300)*x + (10*n^2 + 50*n + 30)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(100 + (30*n - 10)*x + 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-a920b56c9b3ef4d1","steps":["(\u03b8+1)\u00b2"],"edges":["e-ba8060a214ac21f9"],"text":"A038303 \u2192 (\u03b8+1)\u00b2"},"headline":"A038303 \u2192 (\u03b8+1)\u00b2","chain":null}