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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":"c74a7e40f6e202577510f26117b8c1febd603105df4d52e2c425a35cb534430a","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-7de8fc70cf185364","role":"sequence","name":"basis_map_A013611 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, 48, 48;\n 2: 36, 576, 2304, 3072, 1280;\n 3: 216, 5184, 36288, 110592, 161280, 110592, 28672;\n 4: 1296, 41472, 414720, 1990656, 5253120, 7962624, 6881280, 3145728, 589824;\n 5: 7776, 311040, 4043520, 26542080, 102297600, 246841344, 381911040, 377487360, 230031360, 78643200, 11534336;\n 6: 46656, 2239488, 35831808, 298598400, 1517875200, 5064228864, 11494490112, 18006147072, 19428802560, 14155776000, 6643777536, 1811939328, 218103808;\n 7: 279936, 15676416, 297851904, 3009871872, 19020718080, 81266540544, 244765753344, 531351207936, 839196868608, 963158999040, 793516179456, 456608710656, 174046838784, 39460012032, 4026531840;\n 8: 1679616, 107495424, 2364899328, 28092137472, 212363182080, 1107632848896, 4170121740288, 11635368394752, 24416845037568, 38784561315840, 46599455637504, 42008001380352, 27920777084928, 13258564042752, 4252017623040, 824633720832, 73014444032;\n 9: 10077696, 725594112, 18139852800, 247669456896, 2180007198720, 13436068036608, 60791385489408, 207822193164288, 546441378398208, 1116530603458560, 1780994122186752, 2216770060419072, 2140886041362432, 1585345443397632, 881907109724160, 356241767399424, 98569499443200, 16698832846848, 1305670057984;\n","recurrence_sentence":"The row polynomials satisfy (3+12*n*x+(16*n-8)*x^2)*P(n,x) = (18+(72*n+144)*x+(384*n+384)*x^2+(576*n+384)*x^3+(256*n+128)*x^4)*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[((18 + (72*n + 144)*x + (384*n + 384)*x^2 + (576*n + 384)*x^3 + (256*n + 128)*x^4)*P[n - 1, x])/(3 + 12*n*x + (16*n - 8)*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] = ((18 + (72*n + 144)*x + (384*n + 384)*x^2 + (576*n + 384)*x^3 + (256*n + 128)*x^4)*v[i - 1])/(3 + 12*n*x + (16*n - 8)*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-d3c3090977105807","steps":["basis A013611","\u03b8+1"],"edges":["e-bced44278fba7ed1","e-eea1020c6ea2078a"],"text":"A272867 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A272867 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}