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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":"1ece2237b8b411e1693ded06e56f66d98bf5838f4cf37f0f8d45b284f22184cf","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-0990b5be9f344750","role":"sequence","name":"Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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,   8;\n  2: 1,  20,   36;\n  3: 1,  36,  171,    128;\n  4: 1,  56,  495,   1040,     400;\n  5: 1,  80, 1125,   4560,    5275,    1152;\n  6: 1, 108, 2205,  14560,   33775,   23940,     3136;\n  7: 1, 140, 3906,  38080,  147525,  218484,   100891,    8192;\n  8: 1, 176, 6426,  86688,  504525, 1280664,  1290415,  403520,   20736;\n  9: 1, 216, 9990, 178080, 1452675, 5639760, 10006045, 7145280, 1552851, 51200;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(n^2+3*n+2+(4*n^3+12*n^2+12*n)*x-(9*n^4+34*n^3+41*n^2+16*n)*x^2)*P(n-2,x) - (x-(4*n^3+16*n^2+20*n+13)*x^2-(4*n^4+2*n^3-31*n^2-52*n-23)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3+15*n^2+18*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+5*n+3+(4*n^3+12*n^2+12*n+4)*x)*P(n-1,x) + (x-(4*n^3+16*n^2+20*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*P[n - 2, x] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*P[n - 1, x] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*v[i - 2] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*v[i - 1] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n + 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-0990b5be9f344750","steps":["(\u03b8+1)\u00b2"],"edges":["e-e293f6e7721f27fd"],"text":"A126351 \u2192 (\u03b8+1)\u00b2"},"headline":"A126351 \u2192 (\u03b8+1)\u00b2","chain":null}