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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":"4262f7b233ebe55c472d967ab87d04528805b1ecf317dd4c69b887b896be7bc3","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-1639e69c5d7e0691","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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:       3,        2;\n  2:      11,       14,        3;\n  3:      49,       92,       39,        4;\n  4:     257,      644,      417,       92,        5;\n  5:    1539,     4854,     4287,     1512,      205,       6;\n  6:   10299,    39290,    44232,    21752,     4955,     450,       7;\n  7:   75905,   340472,   467466,   296712,    96305,   15444,     987,      8;\n  8:  609441,  3147464,  5111742,  3990584,  1693355,  395628,   46949,   2168,    9;\n  9: 5284451, 30919274, 58083474, 53968736, 28415645, 8798694, 1558403, 141200, 4761, 10;\n","recurrence_sentence":"The row polynomials satisfy -(4*x-n*x^3)*P(n,x) = (64*n-32+(16*n^2+64*n-60)*x-16*x^2-(4*n^3+16*n^2+5*n-15)*x^3-(4*n^3+6*n^2-4*n-6)*x^4-(n^3-n)*x^5)*P(n-2,x) - (64+(32*n+112)*x+(32*n+48)*x^2-(8*n^2+12*n+16)*x^3-(12*n^2+24*n+16)*x^4-(6*n^2+9*n+3)*x^5-(n^2+n)*x^6)*d/dx P(n-2,x) - (32+(16*n+16)*x-8*x^2-(4*n^2+6*n+3)*x^3-2*n^2*x^4)*P(n-1,x) + (16*x+16*x^2-(4*n-4)*x^3-4*n*x^4-n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((64*n - 32 + (16*n^2 + 64*n - 60)*x - 16*x^2 - (4*n^3 + 16*n^2 + 5*n - 15)*x^3 - (4*n^3 + 6*n^2 - 4*n - 6)*x^4 - (n^3 - n)*x^5)*P[n - 2, x] - (64 + (32*n + 112)*x + (32*n + 48)*x^2 - (8*n^2 + 12*n + 16)*x^3 - (12*n^2 + 24*n + 16)*x^4 - (6*n^2 + 9*n + 3)*x^5 - (n^2 + n)*x^6)*D[P[n - 2, x], x] - (32 + (16*n + 16)*x - 8*x^2 - (4*n^2 + 6*n + 3)*x^3 - 2*n^2*x^4)*P[n - 1, x] + (16*x + 16*x^2 - (4*n - 4)*x^3 - 4*n*x^4 - n*x^5)*D[P[n - 1, x], x])/(-4*x + n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((64*n - 32 + (16*n^2 + 64*n - 60)*x - 16*x^2 - (4*n^3 + 16*n^2 + 5*n - 15)*x^3 - (4*n^3 + 6*n^2 - 4*n - 6)*x^4 - (n^3 - n)*x^5)*v[i - 2] - (64 + (32*n + 112)*x + (32*n + 48)*x^2 - (8*n^2 + 12*n + 16)*x^3 - (12*n^2 + 24*n + 16)*x^4 - (6*n^2 + 9*n + 3)*x^5 - (n^2 + n)*x^6)*deriv(v[i - 2], x) - (32 + (16*n + 16)*x - 8*x^2 - (4*n^2 + 6*n + 3)*x^3 - 2*n^2*x^4)*v[i - 1] + (16*x + 16*x^2 - (4*n - 4)*x^3 - 4*n*x^4 - n*x^5)*deriv(v[i - 1], x))/(-4*x + n*x^3)); 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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","x \u21a6 x+1","\u03b8+1"],"edges":["e-236d0fb15dc6eca0","e-8abcc7766de04335","e-8949b75618e516b8"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}