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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":"c5eb40a8e09082c7dbba72091c130d98e1f755a07138caca1ee3d6c29cea208f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2a657d152d421a92","role":"sequence","name":"Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.","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,   6;\n   3: 1,  16,   36;\n   4: 1,  30,  165,    220;\n   5: 1,  48,  468,   1456,    1365;\n   6: 1,  70, 1050,   5600,   11900,    8568;\n   7: 1,  96, 2040,  16320,   58140,   93024,    54264;\n   8: 1, 126, 3591,  39900,  209475,  553014,   706629,   346104;\n   9: 1, 160, 5880,  86240,  619850, 2380224,  4958800,  5262400,  2220075;\n  10: 1, 198, 9108, 170016, 1593900, 8288280, 24864840, 42625440, 38629305, 14307150;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-2*n^2-2*n)*P(n,x) = (4*n^3-2*n^2-2*n+(18*n^3-6*n^2)*x)*P(n-1,x) + ((8*n^2-n-1)*x+(9*n^2-3*n)*x^2)*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[((4*n^3 - 2*n^2 - 2*n + (18*n^3 - 6*n^2)*x)*P[n - 1, x] + ((8*n^2 - n - 1)*x + (9*n^2 - 3*n)*x^2)*D[P[n - 1, x], x])/(4*n^3 - 2*n^2 - 2*n)]];\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] = ((4*n^3 - 2*n^2 - 2*n + (18*n^3 - 6*n^2)*x)*v[i - 1] + ((8*n^2 - n - 1)*x + (9*n^2 - 3*n)*x^2)*deriv(v[i - 1], x))/(4*n^3 - 2*n^2 - 2*n)); 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-2a657d152d421a92","steps":["\u03b8+1"],"edges":["e-3abcbe06e4b32101"],"text":"A102537 \u2192 \u03b8+1"},"headline":"A102537 \u2192 \u03b8+1","chain":null}