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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":"f7512711540efbf5c2963f52db97436044ca99e7ea77a99f8c0bd4bc3e8973aa","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-369f42b9099fbd84","role":"sequence","name":"reversal of Triangular array formed by the Mersenne numbers.","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,  6;\n  2: 1, 12,  21;\n  3: 1, 18,  69,   60;\n  4: 1, 24, 144,  288,   155;\n  5: 1, 30, 246,  792,  1005,    378;\n  6: 1, 36, 375, 1680,  3495,   3132,    889;\n  7: 1, 42, 531, 3060,  8975,  13338,   9037,   2040;\n  8: 1, 48, 714, 5040, 19200,  40608,  45934,  24672,  4599;\n  9: 1, 54, 924, 7728, 36330, 100548, 162932, 146592, 64629, 10230;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(12*n^3+6*n^2)*x+(2*n^3-2*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(12*n^3+42*n^2+6*n-6)*x+(56*n^3+85*n^2-21*n-12)*x^2+(15*n^3+9*n^2-18*n-6)*x^3+(n^3-n)*x^4)*P(n-1,x) - ((2*n+1)*x+(6*n^2+24*n+9)*x^2+(37*n^2+75*n+19)*x^3+(12*n^2+18*n+3)*x^4+(n^2+n)*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[((4*n^2 + 2*n + (12*n^3 + 42*n^2 + 6*n - 6)*x + (56*n^3 + 85*n^2 - 21*n - 12)*x^2 + (15*n^3 + 9*n^2 - 18*n - 6)*x^3 + (n^3 - n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (37*n^2 + 75*n + 19)*x^3 + (12*n^2 + 18*n + 3)*x^4 + (n^2 + n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (2*n^3 - 2*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] = ((4*n^2 + 2*n + (12*n^3 + 42*n^2 + 6*n - 6)*x + (56*n^3 + 85*n^2 - 21*n - 12)*x^2 + (15*n^3 + 9*n^2 - 18*n - 6)*x^3 + (n^3 - n)*x^4)*v[i - 1] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (37*n^2 + 75*n + 19)*x^3 + (12*n^2 + 18*n + 3)*x^4 + (n^2 + n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (2*n^3 - 2*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-9b5c7e97e5191da6","steps":["reverse","\u03b8+1"],"edges":["e-5faf1c18373ba45e","e-d15a5cae4dd7673f"],"text":"A110441 \u2192 reverse \u2192 \u03b8+1"},"headline":"A110441 \u2192 reverse \u2192 \u03b8+1","chain":null}