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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":"721282e67e2b30e4bc4d27ce8fbe74b4748c06e0d7a950c6ea433ad7f71f30a7","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-497753d6192915db","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).","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;\n  2: 1,      2;\n  3: 1,     10;\n  4: 1,     40,      9;\n  5: 1,    168,    105;\n  6: 1,    818,    885,     60;\n  7: 1,   4730,   7077,   1260;\n  8: 1,  32128,  59010,  17920,   525;\n  9: 1, 251328, 531234, 226688, 17325;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+3-(2*n^2-8*n+6)*x+(n^2-4*n+3)*x^2)*P(n-4,x) + (2*n^2-6*n+4-(2*n^2-6*n+4)*x)*P(n-3,x) - (n^2-1-(2*n-2)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 10*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 1 + 10*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 4, x] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*P[n - 3, x] - (n^2 - 1 - (2*n - 2)*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; v[4] = 1 + 10*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*v[i - 4] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*v[i - 3] - (n^2 - 1 - (2*n - 2)*x)*v[i - 2] + (2*n - 1)*v[i - 1]); 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-497753d6192915db","steps":["\u03b8+1"],"edges":["e-c4d209993106d969"],"text":"A136394 \u2192 \u03b8+1"},"headline":"A136394 \u2192 \u03b8+1","chain":null}