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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":"591807018221208b0ee1c0bd29c94ebc8a2af3c634263ae598e7a831d59d821e","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-2eda446fe9bdb6ec","role":"sequence","name":"E2(n, k), the second-order Eulerian numbers with E2(0, k) = \u03b4_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.","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: 0, 4;\n  2: 0, 4,   18;\n  3: 0, 4,   72,      96;\n  4: 0, 4,  198,     928,      600;\n  5: 0, 4,  468,    5248,    11100,      4320;\n  6: 0, 4, 1026,   23232,   110000,    133488,     35280;\n  7: 0, 4, 2160,   89760,   803000,   2093040,   1665216,    322560;\n  8: 0, 4, 4446,  319200,  4895000,  23184720,  38479896,  21832704,   3265920;\n  9: 0, 4, 9036, 1076160, 26562500, 207558000, 609563136, 705682944, 302447520, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-4*n+2)*P(n,x) = ((8*n-12)*x+(72*n^4-260*n^3+316*n^2-140*n+12)*x^2)*P(n-2,x) + (x+(8*n^2-6*n-7)*x^2+(32*n^4-168*n^3+252*n^2-128*n+12)*x^3)*d/dx P(n-2,x) - ((2*n^2-2*n-1)*x^2-(16*n^3-32*n^2+18*n-3)*x^3+(16*n^3-34*n^2+20*n-2)*x^4)*d^2/dx^2 P(n-2,x) - (2*n^2-2*n-(16*n^3-40*n^2+32*n-8)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(16*n^3-32*n^2+16*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((8*n - 12)*x + (72*n^4 - 260*n^3 + 316*n^2 - 140*n + 12)*x^2)*P[n - 2, x] + (x + (8*n^2 - 6*n - 7)*x^2 + (32*n^4 - 168*n^3 + 252*n^2 - 128*n + 12)*x^3)*D[P[n - 2, x], x] - ((2*n^2 - 2*n - 1)*x^2 - (16*n^3 - 32*n^2 + 18*n - 3)*x^3 + (16*n^3 - 34*n^2 + 20*n - 2)*x^4)*D[P[n - 2, x], {x, 2}] - (2*n^2 - 2*n - (16*n^3 - 40*n^2 + 32*n - 8)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 4*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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((8*n - 12)*x + (72*n^4 - 260*n^3 + 316*n^2 - 140*n + 12)*x^2)*v[i - 2] + (x + (8*n^2 - 6*n - 7)*x^2 + (32*n^4 - 168*n^3 + 252*n^2 - 128*n + 12)*x^3)*deriv(v[i - 2], x) - ((2*n^2 - 2*n - 1)*x^2 - (16*n^3 - 32*n^2 + 18*n - 3)*x^3 + (16*n^3 - 34*n^2 + 20*n - 2)*x^4)*deriv(deriv(v[i - 2], x), x) - (2*n^2 - 2*n - (16*n^3 - 40*n^2 + 32*n - 8)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 4*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-e14e37227fef094e","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-c8375d3b3fe52ed3","e-a66389a51d82de08"],"text":"A163936 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A163936 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}