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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":"2de9db7678a0133e1ad490d950709cb92965e110c2cafd717af81c81c46f072f","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-6120ac587df8130e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).","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:   2,     4;\n  2:   4,    20;\n  3:   8,    68,    18;\n  4:  16,   196,   144;\n  5:  32,   516,   702,    64;\n  6:  64,  1284,  2700,   704;\n  7: 128,  3076,  9018,  4480,   200;\n  8: 256,  7172, 27432, 21632,  2800;\n  9: 512, 16388, 78102, 88000, 22000, 576;\n","recurrence_sentence":"The row polynomials satisfy (2+(n+17)*x-(11*n^2+14*n-55)*x^2+(n^3-35*n^2-72*n-16)*x^3+(2*n^4+12*n^3+26*n^2+24*n+8)*x^4)*P(n,x) = -(4+(2*n+32)*x-(22*n^2+74*n-24)*x^2+(2*n^3-42*n^2-206*n-304)*x^3+(4*n^4+38*n^3+212*n^2+518*n+388)*x^4-(4*n^4+40*n^3+148*n^2+240*n+144)*x^5)*P(n-2,x) + (6+(3*n+58)*x-(33*n^2+75*n-204)*x^2+(3*n^3-106*n^2-336*n+40)*x^3+(6*n^4+48*n^3+106*n^2+60*n+16)*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4 + (2*n + 32)*x - (22*n^2 + 74*n - 24)*x^2 + (2*n^3 - 42*n^2 - 206*n - 304)*x^3 + (4*n^4 + 38*n^3 + 212*n^2 + 518*n + 388)*x^4 - (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^5)*P[n - 2, x] + (6 + (3*n + 58)*x - (33*n^2 + 75*n - 204)*x^2 + (3*n^3 - 106*n^2 - 336*n + 40)*x^3 + (6*n^4 + 48*n^3 + 106*n^2 + 60*n + 16)*x^4)*P[n - 1, x])/(2 + (n + 17)*x - (11*n^2 + 14*n - 55)*x^2 + (n^3 - 35*n^2 - 72*n - 16)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4 + (2*n + 32)*x - (22*n^2 + 74*n - 24)*x^2 + (2*n^3 - 42*n^2 - 206*n - 304)*x^3 + (4*n^4 + 38*n^3 + 212*n^2 + 518*n + 388)*x^4 - (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^5)*v[i - 2] + (6 + (3*n + 58)*x - (33*n^2 + 75*n - 204)*x^2 + (3*n^3 - 106*n^2 - 336*n + 40)*x^3 + (6*n^4 + 48*n^3 + 106*n^2 + 60*n + 16)*x^4)*v[i - 1])/(2 + (n + 17)*x - (11*n^2 + 14*n - 55)*x^2 + (n^3 - 35*n^2 - 72*n - 16)*x^3 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^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-6120ac587df8130e","steps":["(\u03b8+1)\u00b2"],"edges":["e-4a419a5baa9e01dc"],"text":"A119808 \u2192 (\u03b8+1)\u00b2"},"headline":"A119808 \u2192 (\u03b8+1)\u00b2","chain":null}