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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":"12c625564698a7f85f2f8a8fd44726915603af72caf9510c2883649b4283ec8e","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-01d7f2ed2a619101","role":"sequence","name":"basis_map_A013611 of Square the entries of Pascal's triangle.","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:     2,      16;\n  2:     6,      96,      144;\n  3:    20,     480,     1728,      1024;\n  4:    70,    2240,    12960,     20480,       6400;\n  5:   252,   10080,    80640,    215040,     192000,      36864;\n  6:   924,   44352,   453600,   1720320,    2688000,    1548288,     200704;\n  7:  3432,  192192,  2395008,  11827200,   26880000,   27869184,   11239424,    1048576;\n  8: 12870,  823680, 12108096,  73801728,  221760000,  340623360,  252887040,   75497472,   5308416;\n  9: 48620, 3500640, 59304960, 430510080, 1614412800, 3321077760, 3709009920, 2076180480, 477757440, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (n^2+(6*n^3-3*n^2)*x+4*n^4*x^2)*P(n,x) = (4*n^2-2*n+(24*n^3+24*n^2-26*n)*x+(16*n^4+112*n^3+8*n^2-88*n+8)*x^2+(48*n^4+96*n^3-16*n^2-64*n)*x^3)*P(n-1,x) - (2*n*x+(12*n^2+20*n-1)*x^2+(8*n^3+72*n^2+64*n-4)*x^3+(32*n^3+96*n^2+64*n)*x^4)*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[((4*n^2 - 2*n + (24*n^3 + 24*n^2 - 26*n)*x + (16*n^4 + 112*n^3 + 8*n^2 - 88*n + 8)*x^2 + (48*n^4 + 96*n^3 - 16*n^2 - 64*n)*x^3)*P[n - 1, x] - (2*n*x + (12*n^2 + 20*n - 1)*x^2 + (8*n^3 + 72*n^2 + 64*n - 4)*x^3 + (32*n^3 + 96*n^2 + 64*n)*x^4)*D[P[n - 1, x], x])/(n^2 + (6*n^3 - 3*n^2)*x + 4*n^4*x^2)]];\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] = ((4*n^2 - 2*n + (24*n^3 + 24*n^2 - 26*n)*x + (16*n^4 + 112*n^3 + 8*n^2 - 88*n + 8)*x^2 + (48*n^4 + 96*n^3 - 16*n^2 - 64*n)*x^3)*v[i - 1] - (2*n*x + (12*n^2 + 20*n - 1)*x^2 + (8*n^3 + 72*n^2 + 64*n - 4)*x^3 + (32*n^3 + 96*n^2 + 64*n)*x^4)*deriv(v[i - 1], x))/(n^2 + (6*n^3 - 3*n^2)*x + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b13531ad79e7c629","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-52e1bd34b019f2ca","e-223a1c61e333f1fc"],"text":"A008459 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A008459 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}