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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":"cb27b7b1acd457a60f65bd8fc99abd6603c815206cc0e610e4914d765f89c3a1","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-e3d901814302c059","role":"sequence","name":"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: 1,  8;\n  3: 1, 16;\n  4: 1, 24,  36;\n  5: 1, 32, 108;\n  6: 1, 40, 216,  128;\n  7: 1, 48, 360,  512;\n  8: 1, 56, 540, 1280,  400;\n  9: 1, 64, 756, 2560, 2000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n+(6*n^3-4*n)*x+(4*n^4+4*n^3)*x^2)*P(n,x) = (n^2+n+(6*n^3+16*n^2-4*n-16)*x+(4*n^4+44*n^3+32*n^2-56*n-16)*x^2+(16*n^4+64*n^3+48*n^2-64*n-64)*x^3)*P(n-1,x) - ((n+1)*x+(6*n^2+20*n+12)*x^2+(4*n^3+68*n^2+128*n+48)*x^3+(32*n^3+160*n^2+256*n+128)*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[((n^2 + n + (6*n^3 + 16*n^2 - 4*n - 16)*x + (4*n^4 + 44*n^3 + 32*n^2 - 56*n - 16)*x^2 + (16*n^4 + 64*n^3 + 48*n^2 - 64*n - 64)*x^3)*P[n - 1, x] - ((n + 1)*x + (6*n^2 + 20*n + 12)*x^2 + (4*n^3 + 68*n^2 + 128*n + 48)*x^3 + (32*n^3 + 160*n^2 + 256*n + 128)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (6*n^3 - 4*n)*x + (4*n^4 + 4*n^3)*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] = ((n^2 + n + (6*n^3 + 16*n^2 - 4*n - 16)*x + (4*n^4 + 44*n^3 + 32*n^2 - 56*n - 16)*x^2 + (16*n^4 + 64*n^3 + 48*n^2 - 64*n - 64)*x^3)*v[i - 1] - ((n + 1)*x + (6*n^2 + 20*n + 12)*x^2 + (4*n^3 + 68*n^2 + 128*n + 48)*x^3 + (32*n^3 + 160*n^2 + 256*n + 128)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (6*n^3 - 4*n)*x + (4*n^4 + 4*n^3)*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":["(\u03b8+1)\u00b2"],"edges":["e-97ccb3d326f9aac4"],"text":"A128099 \u2192 (\u03b8+1)\u00b2"},"headline":"A128099 \u2192 (\u03b8+1)\u00b2","chain":null}