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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":"f287a499d36458332755d383b5462357caffff98f2003792bd19b10e012a1011","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-5e3f8ad0b0a35d3e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 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,     4;\n  3: 1,    20;\n  4: 1,    72,     45;\n  5: 1,   232,    549;\n  6: 1,   716,   4311,     976;\n  7: 1,  2172,  27999,   22160;\n  8: 1,  6544, 164430,  304448,   34625;\n  9: 1, 19664, 910494, 3300416, 1263025;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = (n^2+n-(4*n^3-4*n^2-16*n+16)*x+(9*n^4-13*n^3-10*n^2)*x^2)*P(n-2,x) - ((4*n^2+4*n-6)*x-(4*n^3+20*n^2+20*n-20)*x^2-(4*n^4-32*n^3-18*n^2)*x^3)*d/dx P(n-2,x) - ((4*n^2+4*n-4)*x^2-(8*n^3+8*n^2+4*n-4)*x^3+(8*n^3+4*n^2)*x^4)*d^2/dx^2 P(n-2,x) - (n-(4*n^3-4*n^2)*x)*P(n-1,x) + ((4*n^2+2*n-2)*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n - (4*n^3 - 4*n^2 - 16*n + 16)*x + (9*n^4 - 13*n^3 - 10*n^2)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n - 6)*x - (4*n^3 + 20*n^2 + 20*n - 20)*x^2 - (4*n^4 - 32*n^3 - 18*n^2)*x^3)*D[P[n - 2, x], x] - ((4*n^2 + 4*n - 4)*x^2 - (8*n^3 + 8*n^2 + 4*n - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*D[P[n - 2, x], {x, 2}] - (n - (4*n^3 - 4*n^2)*x)*P[n - 1, x] + ((4*n^2 + 2*n - 2)*x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n - (4*n^3 - 4*n^2 - 16*n + 16)*x + (9*n^4 - 13*n^3 - 10*n^2)*x^2)*v[i - 2] - ((4*n^2 + 4*n - 6)*x - (4*n^3 + 20*n^2 + 20*n - 20)*x^2 - (4*n^4 - 32*n^3 - 18*n^2)*x^3)*deriv(v[i - 2], x) - ((4*n^2 + 4*n - 4)*x^2 - (8*n^3 + 8*n^2 + 4*n - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - (4*n^3 - 4*n^2)*x)*v[i - 1] + ((4*n^2 + 2*n - 2)*x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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-d8312bb744732845","steps":["\u03b3 (scale 1/4)","(\u03b8+1)\u00b2"],"edges":["e-4c46966ead0b055f","e-1aee28e48ce84e67"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 (\u03b8+1)\u00b2"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 (\u03b8+1)\u00b2","chain":null}