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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":"93eece9f328460c5eaed7800356dce96d9e48860f94883ff9a3ab910d80c4b08","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-36098d4730252f73","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    1,      2;\n   3:    2,     12,       3;\n   4:    5,     60,      45,       4;\n   5:   14,    280,     420,     112,       5;\n   6:   42,   1260,    3150,    1680,     225,       6;\n   7:  132,   5544,   20790,   18480,    4950,     396,      7;\n   8:  429,  24024,  126126,  168168,   75075,   12012,    637,     8;\n   9: 1430, 102960,  720720, 1345344,  900900,  240240,  25480,   960,    9;\n  10: 4862, 437580, 3938220, 9801792, 9189180, 3675672, 649740, 48960, 1377, 10;\n","recurrence_sentence":"The row polynomials satisfy -(12*n^2-6*n-(2*n^4-3*n^3+n^2)*x)*P(n,x) = -(48*n^2-96*n+36-(8*n^4-24*n^3-50*n^2+138*n-72)*x-(6*n^4-10*n^3-3*n^2+13*n-6)*x^2)*P(n-1,x) - ((96*n-72)*x-(16*n^3-4*n^2+4*n-6)*x^2+(4*n^3-n^2-5*n+3)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(48*n^2 - 96*n + 36 - (8*n^4 - 24*n^3 - 50*n^2 + 138*n - 72)*x - (6*n^4 - 10*n^3 - 3*n^2 + 13*n - 6)*x^2)*P[n - 1, x] - ((96*n - 72)*x - (16*n^3 - 4*n^2 + 4*n - 6)*x^2 + (4*n^3 - n^2 - 5*n + 3)*x^3)*D[P[n - 1, x], x])/(-12*n^2 + 6*n + (2*n^4 - 3*n^3 + n^2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(48*n^2 - 96*n + 36 - (8*n^4 - 24*n^3 - 50*n^2 + 138*n - 72)*x - (6*n^4 - 10*n^3 - 3*n^2 + 13*n - 6)*x^2)*v[i - 1] - ((96*n - 72)*x - (16*n^3 - 4*n^2 + 4*n - 6)*x^2 + (4*n^3 - n^2 - 5*n + 3)*x^3)*deriv(v[i - 1], x))/(-12*n^2 + 6*n + (2*n^4 - 3*n^3 + n^2)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-36098d4730252f73","steps":["\u03b8+1"],"edges":["e-d889ae30909027d8"],"text":"A280580 \u2192 \u03b8+1"},"headline":"A280580 \u2192 \u03b8+1","chain":null}