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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":"3e525cabd01b1d186f2d0bedde3894b561e1c13f34ffb42cc94458aa01505116","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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,   4;\n   3: 1,  12,   18;\n   4: 1,  24,   90,    80;\n   5: 1,  40,  270,   560,    350;\n   6: 1,  60,  630,  2240,   3150,    1512;\n   7: 1,  84, 1260,  6720,  15750,   16632,    6468;\n   8: 1, 112, 2268, 16800,  57750,   99792,   84084,   27456;\n   9: 1, 144, 3780, 36960, 173250,  432432,  588588,  411840,  115830;\n  10: 1, 180, 5940, 73920, 450450, 1513512, 2942940, 3294720, 1969110, 486200;\n","recurrence_sentence":"The row polynomials satisfy -(n-1-(n^3-4*n^2+5*n-2)*x)*P(n,x) = -(n-1-(n^3-4*n^2+n+2)*x-(2*n^3-6*n^2+4*n)*x^2)*P(n-1,x) - (2*x-(2*n^2-4*n-2)*x^2-(2*n^2-4*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 - (n^3 - 4*n^2 + n + 2)*x - (2*n^3 - 6*n^2 + 4*n)*x^2)*P[n - 1, x] - (2*x - (2*n^2 - 4*n - 2)*x^2 - (2*n^2 - 4*n)*x^3)*D[P[n - 1, x], x])/(-n + 1 + (n^3 - 4*n^2 + 5*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 1 - (n^3 - 4*n^2 + n + 2)*x - (2*n^3 - 6*n^2 + 4*n)*x^2)*v[i - 1] - (2*x - (2*n^2 - 4*n - 2)*x^2 - (2*n^2 - 4*n)*x^3)*deriv(v[i - 1], x))/(-n + 1 + (n^3 - 4*n^2 + 5*n - 2)*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-4689fb1386d440ad","steps":["\u03b8+1"],"edges":["e-4000f98bcecb476b"],"text":"A063007 \u2192 \u03b8+1"},"headline":"A063007 \u2192 \u03b8+1","chain":null}