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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":"eac26cdb8cc471094864a197ddbc0dfae1e9070de0d214e5f71abac395301856","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-7e353d0275e3ebc5","role":"sequence","name":"Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.","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: 0,     2;\n  1: 0,     6,     12,       4;\n  2: 0,    18,     72,      88,       40,        6;\n  3: 0,    54,    324,     684,      680,      342,        84,         8;\n  4: 0,   162,   1296,    3888,     6000,     5316,      2800,       864,       144,        10;\n  5: 0,   486,   4860,   18900,    39600,    50580,     41608,     22480,      7920,      1750,       220,       12;\n  6: 0,  1458,  17496,   83592,   221400,   370170,    416304,    325088,    178416,     68550,     18040,     3096,      312,       14;\n  7: 0,  4374,  61236,  347004,  1111320,  2298618,   3286332,   3371928,   2533392,   1404970,    573804,   170268,    35672,     4998,     420,     16;\n  8: 0, 13122, 209952, 1376352,  5171040, 12750696,  22141728,  28203840,  26999136,  19689340,  10999648,  4700640,  1522976,   367304,   63840,   7552,    544,    18;\n  9: 0, 39366, 708588, 5275044, 22744800, 65137608, 133167888, 202831776, 236382624, 214318980, 152630984, 85727592, 37937952, 13146504, 3522960, 714816, 106080, 10854, 684, 20;\n","recurrence_sentence":"The row polynomials satisfy (3+(2*n+2)*x+n*x^2)*P(n,x) = (9+(6*n+24)*x+(11*n+22)*x^2+(6*n+8)*x^3+(n+1)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 2*x.","mathematica":"P[0, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9 + (6*n + 24)*x + (11*n + 22)*x^2 + (6*n + 8)*x^3 + (n + 1)*x^4)*P[n - 1, x])/(3 + (2*n + 2)*x + n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (6*n + 24)*x + (11*n + 22)*x^2 + (6*n + 8)*x^3 + (n + 1)*x^4)*v[i - 1])/(3 + (2*n + 2)*x + n*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-7e353d0275e3ebc5","steps":["\u03b8+1"],"edges":["e-75931ce81f663e9a"],"text":"A299989 \u2192 \u03b8+1"},"headline":"A299989 \u2192 \u03b8+1","chain":null}