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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":"b53aa3a45cec600941b928f72709c645e5c4177ef8d30962429c02060250e938","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-05516f61aeaea779","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","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,       2;\n  2: 1,      24,          3;\n  3: 1,     166,        249,           4;\n  4: 1,    1028,       5478,        2056,            5;\n  5: 1,    6210,      84930,      113240,        15525,            6;\n  6: 1,   37312,    1129845,     3623680,      1883075,       111936,           7;\n  7: 1,  223934,   13883463,    89614540,    112018175,     27766926,      783769,          8;\n  8: 1, 1343676,  163131024,  1915126024,   4704763350,   2872689036,   380639056,    5374704,        9;\n  9: 1, 8062138, 1866999768, 37384765376, 161041626130, 193249951356, 65423339408, 4978666048, 36279621, 10;\n","recurrence_sentence":"The row polynomials satisfy (5*n-4)*P(n,x) = (20*n+4+(200*n^2-320*n-72)*x-(125*n^3-300*n^2+115*n+36)*x^2)*P(n-2,x) + ((100*n+20)*x-(125*n^2+25*n)*x^2+(125*n^2-75*n-20)*x^3)*d/dx P(n-2,x) - (15*n+8-(50*n^2-80*n+32)*x)*P(n-1,x) + ((25*n-20)*x-(25*n-20)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((20*n + 4 + (200*n^2 - 320*n - 72)*x - (125*n^3 - 300*n^2 + 115*n + 36)*x^2)*P[n - 2, x] + ((100*n + 20)*x - (125*n^2 + 25*n)*x^2 + (125*n^2 - 75*n - 20)*x^3)*D[P[n - 2, x], x] - (15*n + 8 - (50*n^2 - 80*n + 32)*x)*P[n - 1, x] + ((25*n - 20)*x - (25*n - 20)*x^2)*D[P[n - 1, x], x])/(5*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 4 + (200*n^2 - 320*n - 72)*x - (125*n^3 - 300*n^2 + 115*n + 36)*x^2)*v[i - 2] + ((100*n + 20)*x - (125*n^2 + 25*n)*x^2 + (125*n^2 - 75*n - 20)*x^3)*deriv(v[i - 2], x) - (15*n + 8 - (50*n^2 - 80*n + 32)*x)*v[i - 1] + ((25*n - 20)*x - (25*n - 20)*x^2)*deriv(v[i - 1], x))/(5*n - 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-05516f61aeaea779","steps":["\u03b8+1"],"edges":["e-35619d5ea7b8539d"],"text":"A142460 \u2192 \u03b8+1"},"headline":"A142460 \u2192 \u03b8+1","chain":null}