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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":"46a3007f73bcb43dbbf0c6ceb4bde2a119e6ac3fc4c906092d5dbe0a023f7aa1","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-351cbe3ca6567d98","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","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: 6, 24, 9;\n 2: 36, 288, 432, 192, 25;\n 3: 216, 2592, 6804, 6912, 3150, 648, 49;\n 4: 1296, 20736, 77760, 124416, 102600, 46656, 11760, 1536, 81;\n 5: 7776, 155520, 758160, 1658880, 1998000, 1446336, 652680, 184320, 31590, 3000, 121;\n 6: 46656, 1119744, 6718464, 18662400, 29646000, 29673216, 19643904, 8792064, 2668140, 540000, 69696, 5184, 169;\n 7: 279936, 7838208, 55847232, 188116992, 371498400, 476171136, 418300848, 259448832, 115246152, 36741600, 8324316, 1306368, 134862, 8232, 225;\n 8: 1679616, 53747712, 443418624, 1755758592, 4147718400, 6490036224, 7126672896, 5681332224, 3353143392, 1479513600, 488847744, 120185856, 21634704, 2765952, 237600, 12288, 289;\n 9: 10077696, 362797056, 3401222400, 15479341056, 42578265600, 78726961152, 103891527936, 101475680256, 75042303552, 42592262400, 18683371872, 6342230016, 1658887776, 330728832, 49280400, 5308416, 390150, 17496, 361;\n","recurrence_sentence":"The row polynomials satisfy (36+(108*n-36)*x+(36*n^2+48*n-36)*x^2+(24*n^2-6*n-6)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (216+(648*n+648)*x+(216*n^2+1368*n+756)*x^2+(360*n^2+1080*n+432)*x^3+(204*n^2+396*n+126)*x^4+(48*n^2+66*n+18)*x^5+(4*n^2+4*n+1)*x^6)*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[((216 + (648*n + 648)*x + (216*n^2 + 1368*n + 756)*x^2 + (360*n^2 + 1080*n + 432)*x^3 + (204*n^2 + 396*n + 126)*x^4 + (48*n^2 + 66*n + 18)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(36 + (108*n - 36)*x + (36*n^2 + 48*n - 36)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (4*n^2 - 4*n + 1)*x^4)]];\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] = ((216 + (648*n + 648)*x + (216*n^2 + 1368*n + 756)*x^2 + (360*n^2 + 1080*n + 432)*x^3 + (204*n^2 + 396*n + 126)*x^4 + (48*n^2 + 66*n + 18)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(36 + (108*n - 36)*x + (36*n^2 + 48*n - 36)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (4*n^2 - 4*n + 1)*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-d3c3090977105807","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-33933faab90bc4be","e-6068a754e6fd6a29"],"text":"A272867 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A272867 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}