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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":"4b343bf784be970e06cfe4b91ea44f660bba91db812a69d335643e419e64a8ad","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-af7a69a897cce4e9","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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;\n  2:   3,   16;\n  3:   5,   32;\n  4:  11,  112,   192;\n  5:  21,  256,   576;\n  6:  43,  656,  2304,   2048;\n  7:  85, 1504,  6528,   8192;\n  8: 171, 3504, 19008,  36864,  20480;\n  9: 341, 7872, 50112, 122880, 102400;\n","recurrence_sentence":"The row polynomials satisfy (72*n+(144*n^2+208*n)*x+(128*n^3+128*n^2)*x^2)*P(n,x) = (360*n-288+(1008*n^2+1904*n-2560)*x+(640*n^3+4736*n^2+1536*n-6656)*x^2+(2048*n^3+4096*n^2-2048*n-4096)*x^3)*P(n-1,x) - (81+(324*n+1080)*x+(288*n^2+3312*n+5264)*x^2+(2176*n^2+11136*n+11008)*x^3+(4096*n^2+12288*n+8192)*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[((360*n - 288 + (1008*n^2 + 1904*n - 2560)*x + (640*n^3 + 4736*n^2 + 1536*n - 6656)*x^2 + (2048*n^3 + 4096*n^2 - 2048*n - 4096)*x^3)*P[n - 1, x] - (81 + (324*n + 1080)*x + (288*n^2 + 3312*n + 5264)*x^2 + (2176*n^2 + 11136*n + 11008)*x^3 + (4096*n^2 + 12288*n + 8192)*x^4)*D[P[n - 1, x], x])/(72*n + (144*n^2 + 208*n)*x + (128*n^3 + 128*n^2)*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] = ((360*n - 288 + (1008*n^2 + 1904*n - 2560)*x + (640*n^3 + 4736*n^2 + 1536*n - 6656)*x^2 + (2048*n^3 + 4096*n^2 - 2048*n - 4096)*x^3)*v[i - 1] - (81 + (324*n + 1080)*x + (288*n^2 + 3312*n + 5264)*x^2 + (2176*n^2 + 11136*n + 11008)*x^3 + (4096*n^2 + 12288*n + 8192)*x^4)*deriv(v[i - 1], x))/(72*n + (144*n^2 + 208*n)*x + (128*n^3 + 128*n^2)*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-e3d901814302c059","steps":["basis A013611","\u03b8+1"],"edges":["e-099e290d18ba9894","e-9e1a7b67c6b50875"],"text":"A128099 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A128099 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}