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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":"aad6ab1745dd45d1a2261b8f26e5c160bd5a53b34b2f29fbbc58738828e59500","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-0c97d42b0cd05e48","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 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: 1,      4;\n  2: 1,     24,         9;\n  3: 1,    108,       189,         16;\n  4: 1,    448,      2430,        928,          25;\n  5: 1,   1812,     25902,      27808,        3525,         36;\n  6: 1,   7272,    253503,     629120,      218475,      11448,        49;\n  7: 1,  29116,   2379699,   12319280,     9393875,    1355940,     33565,       64;\n  8: 1, 116496,  21875868,  222491936,   331091250,  102922560,   7181440,    91776,     81;\n  9: 1, 466020, 198979740, 3832104000, 10363143150, 6089069160, 912246720, 34013184, 238869, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9*n^2-2*n-7)*x^2*P(n-2,x) + ((9*n+7)*x^2+(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) + ((17*n+13)*x^3-(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (4*n+3)*x^4*d^3/dx^3 P(n-2,x) + (1+4*n*x)*P(n-1,x) + (3*x-(4*n+4)*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(9*n^2 - 2*n - 7)*x^2*P[n - 2, x] + ((9*n + 7)*x^2 + (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] + ((17*n + 13)*x^3 - (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (4*n + 3)*x^4*D[P[n - 2, x], {x, 3}] + (1 + 4*n*x)*P[n - 1, x] + (3*x - (4*n + 4)*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (9*n^2 - 2*n - 7)*x^2*v[i - 2] + ((9*n + 7)*x^2 + (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) + ((17*n + 13)*x^3 - (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (4*n + 3)*x^4*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*n*x)*v[i - 1] + (3*x - (4*n + 4)*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-0c97d42b0cd05e48","steps":["(\u03b8+1)\u00b2"],"edges":["e-2735d0cd05a71baa"],"text":"A166960 \u2192 (\u03b8+1)\u00b2"},"headline":"A166960 \u2192 (\u03b8+1)\u00b2","chain":null}