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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":"d196fd8bd9d66defbbc61b63c77e7778229cbfc8dbac6548c83144a658123117","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-01cea33b6485d9cc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","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,      8;\n  2:     9,     48,      36;\n  3:    27,    216,     324,     128;\n  4:    81,    864,    1944,    1536,      400;\n  5:   243,   3240,    9720,   11520,     6000,     1152;\n  6:   729,  11664,   43740,   69120,    54000,    20736,    3136;\n  7:  2187,  40824,  183708,  362880,   378000,   217728,   65856,    8192;\n  8:  6561, 139968,  734832, 1741824,  2268000,  1741824,  790272,  196608,  20736;\n  9: 19683, 472392, 2834352, 7838208, 12247200, 11757312, 7112448, 2654208, 559872, 51200;\n","recurrence_sentence":"The row polynomials satisfy (9+(18*n-6)*x+4*n^2*x^2)*P(n,x) = (27+(54*n+54)*x+(12*n^2+60*n+36)*x^2+(8*n^2+16*n+8)*x^3)*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[((27 + (54*n + 54)*x + (12*n^2 + 60*n + 36)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(9 + (18*n - 6)*x + 4*n^2*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] = ((27 + (54*n + 54)*x + (12*n^2 + 60*n + 36)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(9 + (18*n - 6)*x + 4*n^2*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-01cea33b6485d9cc","steps":["(\u03b8+1)\u00b2"],"edges":["e-635f90021d26fb2c"],"text":"A038220 \u2192 (\u03b8+1)\u00b2"},"headline":"A038220 \u2192 (\u03b8+1)\u00b2","chain":null}