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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":"897b8661b8e300603ba74ba6a3cb611f191d8c394494dffe130fca308a157dbc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5b857304d1bafe7a","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    2;\n   3: 1,    8,      9;\n   4: 1,   22,     75,      60;\n   5: 1,   52,    390,     840,      525;\n   6: 1,  114,   1638,    7000,    11025,      5670;\n   7: 1,  240,   6111,   45472,   133875,    166320,     72765;\n   8: 1,  494,  21213,  255220,  1237775,   2765070,   2837835,    1081080;\n   9: 1, 1004,  70308, 1303720,  9698150,  34414380,  61801740,   54054000,   18243225;\n  10: 1, 2026, 225984, 6246064, 68046550, 357777420, 994713720, 1491890400, 1137161025, 344594250;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+4)*x^2*P(n-2,x) - ((n-2)*x^2+(n-2)*x^3)*d/dx P(n-2,x) + (1+(2*n-2)*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4)*x^2*P[n - 2, x] - ((n - 2)*x^2 + (n - 2)*x^3)*D[P[n - 2, x], x] + (1 + (2*n - 2)*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 4*n + 4)*x^2*v[i - 2] - ((n - 2)*x^2 + (n - 2)*x^3)*deriv(v[i - 2], x) + (1 + (2*n - 2)*x)*v[i - 1] + (x + x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5b857304d1bafe7a","steps":["\u03b8+1"],"edges":["e-2022e84f72fe50a2"],"text":"A112493 \u2192 \u03b8+1"},"headline":"A112493 \u2192 \u03b8+1","chain":null}