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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":"a48a18359ee30a3dee4248e4842771adf3bda763fb766d3740165785159ae584","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-dfe248045567f938","role":"sequence","name":"basis_map_A013609 of 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:       3,        8;\n   3:      13,       72,        72;\n   4:      63,      528,      1080,        640;\n   5:     321,     3600,     11160,      13440,       5600;\n   6:    1683,    23640,     98280,     179200,     151200,      48384;\n   7:    8989,   151704,    791280,    1935360,    2469600,    1596672,     413952;\n   8:   48639,   958496,   6014736,   18466560,   31600800,   30868992,   16144128,    3514368;\n   9:  265729,  5988384,  43923600,  162328320,  348717600,  456648192,  360552192,  158146560,   29652480;\n  10: 1462563, 37097640, 311422320, 1345935360, 3481077600, 5721940224, 6054048000, 4006379520, 1512276480, 248934400;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-2+(3*n-3)*x-(2*n^3-8*n^2+10*n-4)*x^2)*P(n,x) = -(6*n-6+(4*n^2+13*n-17)*x-(6*n^3-24*n^2+6*n+12)*x^2-(8*n^3-24*n^2+16*n)*x^3)*P(n-1,x) - (2+12*x-(4*n^2-8*n-22)*x^2-(12*n^2-24*n-12)*x^3-(8*n^2-16*n)*x^4)*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[(-(6*n - 6 + (4*n^2 + 13*n - 17)*x - (6*n^3 - 24*n^2 + 6*n + 12)*x^2 - (8*n^3 - 24*n^2 + 16*n)*x^3)*P[n - 1, x] - (2 + 12*x - (4*n^2 - 8*n - 22)*x^2 - (12*n^2 - 24*n - 12)*x^3 - (8*n^2 - 16*n)*x^4)*D[P[n - 1, x], x])/(-2*n + 2 - (3*n - 3)*x + (2*n^3 - 8*n^2 + 10*n - 4)*x^2)]];\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] = (-(6*n - 6 + (4*n^2 + 13*n - 17)*x - (6*n^3 - 24*n^2 + 6*n + 12)*x^2 - (8*n^3 - 24*n^2 + 16*n)*x^3)*v[i - 1] - (2 + 12*x - (4*n^2 - 8*n - 22)*x^2 - (12*n^2 - 24*n - 12)*x^3 - (8*n^2 - 16*n)*x^4)*deriv(v[i - 1], x))/(-2*n + 2 - (3*n - 3)*x + (2*n^3 - 8*n^2 + 10*n - 4)*x^2)); 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":["basis A013609","\u03b8+1"],"edges":["e-7c694c7111620c7d","e-dfd7dff686bb59de"],"text":"A063007 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A063007 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}