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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":"5f4d6abd51547bafc6abadfd45905035b6e5f9649aedac82659604aa15f21415","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-f712c326ca8e5415","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.","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: 1,    4,      3;\n  2: 1,   12,     36,      40,        15;\n  3: 1,   28,    183,     496,       655,        420,        105;\n  4: 1,   60,    720,    3560,      9150,      13356,      11200,        5040,          945;\n  5: 1,  124,   2523,   20240,     84950,     211296,     329434,      326368,       200025,        69300,        10395;\n  6: 1,  252,   8316,  101640,    639765,    2408616,    5858832,     9572256,     10639755,      7957180,      3839220,      1081080,       135135;\n  7: 1,  508,  26463,  474096,   4256805,   22738716,   79210803,   189588288,    320787891,    388089460,    334326685,    200600400,     79774695,     18918900,      2027025;\n  8: 1, 1020,  82440, 2110120,  26157300,  190358532,  902615280,  2965988880,   7012577430,  12192535300,  15744228456,  15088118760,  10598928340,   5306961660,   1794592800,    367567200,    34459425;\n  9: 1, 2044, 253443, 9104080, 152299900, 1468699512, 9166427508, 39692623776, 124624534230, 291836790400, 518560641346, 705229331664, 734752126108, 582301119400, 345111666900, 148219431360, 43591172625, 7856748900, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x + 3*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];\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] = (1 + 4*x + 3*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], x)); 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-f712c326ca8e5415","steps":["\u03b8+1"],"edges":["e-6c9a4759e3e35b35"],"text":"A094262 \u2192 \u03b8+1"},"headline":"A094262 \u2192 \u03b8+1","chain":null}