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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":"75f1b6db231cf119476eeea6bbacad88875c30f80536f59466a44c4b6f05c485","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-83e32d0bc1c10d45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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:      3,       2;\n  2:     11,      14,       3;\n  3:     45,      84,      36,       4;\n  4:    197,     484,     312,      72,       5;\n  5:    903,    2750,    2370,     840,     125,      6;\n  6:   4279,   15546,   16785,    8060,    1875,    198,      7;\n  7:  20793,   87752,  113904,   69160,   22050,   3696,    294,     8;\n  8: 103049,  495368,  751296,  552160,  224420,  52080,   6664,   416,   9;\n  9: 518859, 2798586, 4856868, 4192608, 2075220, 618408, 110544, 11232, 567, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n+8+(4*n^2+12*n+8)*x+(n^3+3*n^2+2*n)*x^2)*P(n,x) = (28*n+8+(32*n^2+56*n+8)*x+(7*n^3+33*n^2+26*n)*x^2+(3*n^3+7*n^2+2*n)*x^3)*P(n-1,x) - (8+(16*n+28)*x+(4*n^2+34*n+32)*x^2+(6*n^2+23*n+14)*x^3+(2*n^2+5*n+2)*x^4)*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[((28*n + 8 + (32*n^2 + 56*n + 8)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (3*n^3 + 7*n^2 + 2*n)*x^3)*P[n - 1, x] - (8 + (16*n + 28)*x + (4*n^2 + 34*n + 32)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (2*n^2 + 5*n + 2)*x^4)*D[P[n - 1, x], x])/(4*n + 8 + (4*n^2 + 12*n + 8)*x + (n^3 + 3*n^2 + 2*n)*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] = ((28*n + 8 + (32*n^2 + 56*n + 8)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (3*n^3 + 7*n^2 + 2*n)*x^3)*v[i - 1] - (8 + (16*n + 28)*x + (4*n^2 + 34*n + 32)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (2*n^2 + 5*n + 2)*x^4)*deriv(v[i - 1], x))/(4*n + 8 + (4*n^2 + 12*n + 8)*x + (n^3 + 3*n^2 + 2*n)*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-57b4b14604f4c0d7","steps":["basis A007318","\u03b8+1"],"edges":["e-3c8d43f1dfa996d6","e-7964f96af6e073f8"],"text":"A126216 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A126216 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}