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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":"e145f50587c4f9bb115734be1ee21dc47eb5be8be9a426ee16625d15c8e2a2c3","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-41bbb784f9572762","role":"sequence","name":"theta_plus_one of Number triangle T(n,k)=C(n,k)C(2,n-k).","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,  8;\n  2: 1, 32,  27;\n  3: 0, 24, 162,  64;\n  4: 0,  0, 162, 512,  125;\n  5: 0,  0,   0, 640, 1250,  216;\n  6: 0,  0,   0,   0, 1875, 2592,  343;\n  7: 0,  0,   0,   0,    0, 4536, 4802,   512;\n  8: 0,  0,   0,   0,    0,    0, 9604,  8192,   729;\n  9: 0,  0,   0,   0,    0,    0,    0, 18432, 13122, 1000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^6*P(n-6,x) + 6*x^5*P(n-5,x) - 15*x^4*P(n-4,x) + 20*x^3*P(n-3,x) - 15*x^2*P(n-2,x) + 6*x*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2 + 8*x, P(2,x) = 1 + 32*x + 27*x^2, P(3,x) = 24*x + 162*x^2 + 64*x^3, P(4,x) = 162*x^2 + 512*x^3 + 125*x^4, P(5,x) = 640*x^3 + 1250*x^4 + 216*x^5.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[2, x_] := 1 + 32*x + 27*x^2;\nP[3, x_] := 24*x + 162*x^2 + 64*x^3;\nP[4, x_] := 162*x^2 + 512*x^3 + 125*x^4;\nP[5, x_] := 640*x^3 + 1250*x^4 + 216*x^5;\nP[n_, x_] := P[n, x] = Expand[-x^6*P[n - 6, x] + 6*x^5*P[n - 5, x] - 15*x^4*P[n - 4, x] + 20*x^3*P[n - 3, x] - 15*x^2*P[n - 2, x] + 6*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; v[3] = 1 + 32*x + 27*x^2; v[4] = 24*x + 162*x^2 + 64*x^3; v[5] = 162*x^2 + 512*x^3 + 125*x^4; v[6] = 640*x^3 + 1250*x^4 + 216*x^5; for(n = 6, N, my(i = n + 1); v[i] = -x^6*v[i - 6] + 6*x^5*v[i - 5] - 15*x^4*v[i - 4] + 20*x^3*v[i - 3] - 15*x^2*v[i - 2] + 6*x*v[i - 1]); 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-1815d3e1c9c57d5b","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-83c376d42fc7c950","e-d6726fe8c50332ed"],"text":"A106246 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A106246 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}