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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":"f9139afd1cd343c6c5b8b15f07fd819d887ed01239dae6173f9b8d61d894d691","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d6b7687ba0ff59dd","role":"sequence","name":"Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:   3,    2;\n  2:   6,   12,     3;\n  3:  12,   42,    27,     4;\n  4:  24,  120,   135,    48,     5;\n  5:  48,  312,   513,   312,    75,     6;\n  6:  96,  768,  1674,  1488,   600,   108,     7;\n  7: 192, 1824,  4968,  5892,  3450,  1026,   147,    8;\n  8: 384, 4224, 13824, 20640, 16125,  6912,  1617,  192,   9;\n  9: 768, 9600, 36720, 66336, 65415, 37314, 12495, 2400, 243, 10;\n","recurrence_sentence":"The row polynomials satisfy -(8*n^2-8*n+(6*n^3-16*n^2-6*n)*x-(3*n^3+8*n^2-3*n)*x^2+(n^2+n)*x^3)*P(n,x) = -(16*n^2-16*n+(12*n^3+16*n^2-76*n)*x+(12*n^3-28*n^2-60*n+20)*x^2-(13*n^2+7*n-10)*x^3+(2*n+2)*x^4)*P(n-1,x) + (16*n*x+(12*n^2+44*n-8)*x^2+(24*n^2+24*n-20)*x^3+(3*n^2-5*n-10)*x^4-(n+1)*x^5)*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[Together[(-(16*n^2 - 16*n + (12*n^3 + 16*n^2 - 76*n)*x + (12*n^3 - 28*n^2 - 60*n + 20)*x^2 - (13*n^2 + 7*n - 10)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (16*n*x + (12*n^2 + 44*n - 8)*x^2 + (24*n^2 + 24*n - 20)*x^3 + (3*n^2 - 5*n - 10)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(-8*n^2 + 8*n - (6*n^3 - 16*n^2 - 6*n)*x + (3*n^3 + 8*n^2 - 3*n)*x^2 - (n^2 + n)*x^3)]];\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] = (-(16*n^2 - 16*n + (12*n^3 + 16*n^2 - 76*n)*x + (12*n^3 - 28*n^2 - 60*n + 20)*x^2 - (13*n^2 + 7*n - 10)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (16*n*x + (12*n^2 + 44*n - 8)*x^2 + (24*n^2 + 24*n - 20)*x^3 + (3*n^2 - 5*n - 10)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(-8*n^2 + 8*n - (6*n^3 - 16*n^2 - 6*n)*x + (3*n^3 + 8*n^2 - 3*n)*x^2 - (n^2 + n)*x^3)); 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-d6b7687ba0ff59dd","steps":["\u03b8+1"],"edges":["e-0cc329e6d69bfe0a"],"text":"A116412 \u2192 \u03b8+1"},"headline":"A116412 \u2192 \u03b8+1","chain":null}