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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":"7150d9d38d05354401bd62ba9339a4f53c32c14d50ca6f9bd6b46c270e37aec7","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-991b9bffcf8ddedb","role":"sequence","name":"Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","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,   32;\n  4: 1,   88,    144;\n  5: 1,  208,   1224;\n  6: 1,  456,   6480,    4352;\n  7: 1,  960,  27648,   63488;\n  8: 1, 1976, 104544,  548864,  198400;\n  9: 1, 4016, 366984, 3682304, 4422400;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -((8*n^2-8*n-16)*x-(36*n^3-52*n^2-40*n)*x^2)*P(n-2,x) - ((2*n+2)*x-(8*n^2+24*n+20)*x^2-(16*n^3-128*n^2-72*n)*x^3)*d/dx P(n-2,x) - ((n+1)*x^2-(8*n^2+8*n+4)*x^3+(32*n^2+16*n)*x^4)*d^2/dx^2 P(n-2,x) + (n+(8*n^2-8*n)*x)*P(n-1,x) + ((2*n+1)*x-(8*n^2+8*n)*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[(-((8*n^2 - 8*n - 16)*x - (36*n^3 - 52*n^2 - 40*n)*x^2)*P[n - 2, x] - ((2*n + 2)*x - (8*n^2 + 24*n + 20)*x^2 - (16*n^3 - 128*n^2 - 72*n)*x^3)*D[P[n - 2, x], x] - ((n + 1)*x^2 - (8*n^2 + 8*n + 4)*x^3 + (32*n^2 + 16*n)*x^4)*D[P[n - 2, x], {x, 2}] + (n + (8*n^2 - 8*n)*x)*P[n - 1, x] + ((2*n + 1)*x - (8*n^2 + 8*n)*x^2)*D[P[n - 1, x], x])/(n)]];\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] = (-((8*n^2 - 8*n - 16)*x - (36*n^3 - 52*n^2 - 40*n)*x^2)*v[i - 2] - ((2*n + 2)*x - (8*n^2 + 24*n + 20)*x^2 - (16*n^3 - 128*n^2 - 72*n)*x^3)*deriv(v[i - 2], x) - ((n + 1)*x^2 - (8*n^2 + 8*n + 4)*x^3 + (32*n^2 + 16*n)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + (8*n^2 - 8*n)*x)*v[i - 1] + ((2*n + 1)*x - (8*n^2 + 8*n)*x^2)*deriv(v[i - 1], x))/(n)); 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-ff270bcb802a85af","steps":["\u03b3 (shift +1)","(\u03b8+1)\u00b2"],"edges":["e-e49ea542bfed6bbf","e-019ad329539bbbb3"],"text":"A008292 \u2192 \u03b3 (shift +1) \u2192 (\u03b8+1)\u00b2"},"headline":"A008292 \u2192 \u03b3 (shift +1) \u2192 (\u03b8+1)\u00b2","chain":null}