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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"acb065dfa3e068a544ae0945a8804175ce882721b4acde4d4a545dc6b87f4853","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-20a6b63b1c7531be","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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:         9,          8;\n  2:        81,        144,         36;\n  3:       729,       1944,        972,        128;\n  4:      6561,      23328,      17496,       4608,        400;\n  5:     59049,     262440,     262440,     103680,      18000,      1152;\n  6:    531441,    2834352,    3542940,    1866240,     486000,     62208,      3136;\n  7:   4782969,   29760696,   44641044,   29393280,   10206000,   1959552,    197568,     8192;\n  8:  43046721,  306110016,  535692528,  423263232,  183708000,  47029248,   7112448,   589824,   20736;\n  9: 387420489, 3099363912, 6198727824, 5714053632, 2976069600, 952342272, 192036096, 23887872, 1679616, 51200;\n","recurrence_sentence":"The row polynomials satisfy (81+(54*n-18)*x+4*n^2*x^2)*P(n,x) = (729+(486*n+486)*x+(36*n^2+180*n+108)*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[((729 + (486*n + 486)*x + (36*n^2 + 180*n + 108)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(81 + (54*n - 18)*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] = ((729 + (486*n + 486)*x + (36*n^2 + 180*n + 108)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(81 + (54*n - 18)*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-366b2525683554ec","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-0da8e8e5bf86fd8d","e-c79ea04a16bfad1b"],"text":"A038279 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038279 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}