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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":"7251e356748357f0e265215009270095ba2f0a791216773201b2eef59c09a084","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6bdb5d7e28f56300","role":"sequence","name":"basis_map_A007318 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:     4,      4;\n  2:    14,     32,       9;\n  3:    48,    176,     108,      16;\n  4:   164,    832,     810,     256,      25;\n  5:   560,   3632,    4896,    2432,     500,     36;\n  6:  1912,  15104,   26136,   17920,    5750,    864,     49;\n  7:  6528,  60800,  128736,  113280,   50000,  11664,   1372,    64;\n  8: 22288, 239104,  599472,  646144,  365500, 116928,  21266,  2048,   81;\n  9: 76096, 923968, 2678400, 3423744, 2371600, 970704, 241472, 35840, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (512*n+64*n*x-(160*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (1024*n+1024+(448*n^2+1152*n+704)*x-(320*n^3+352*n^2-192*n-224)*x^2-(16*n^4+216*n^3+336*n^2+120*n-16)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (512+768*x-(448*n^2+448*n-320)*x^2-(608*n^2+608*n)*x^3+(8*n^4+16*n^3-208*n^2-216*n-32)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+n^2)*x^6)*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[((1024*n + 1024 + (448*n^2 + 1152*n + 704)*x - (320*n^3 + 352*n^2 - 192*n - 224)*x^2 - (16*n^4 + 216*n^3 + 336*n^2 + 120*n - 16)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (512 + 768*x - (448*n^2 + 448*n - 320)*x^2 - (608*n^2 + 608*n)*x^3 + (8*n^4 + 16*n^3 - 208*n^2 - 216*n - 32)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(512*n + 64*n*x - (160*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)]];\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] = ((1024*n + 1024 + (448*n^2 + 1152*n + 704)*x - (320*n^3 + 352*n^2 - 192*n - 224)*x^2 - (16*n^4 + 216*n^3 + 336*n^2 + 120*n - 16)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (512 + 768*x - (448*n^2 + 448*n - 320)*x^2 - (608*n^2 + 608*n)*x^3 + (8*n^4 + 16*n^3 - 208*n^2 - 216*n - 32)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(512*n + 64*n*x - (160*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)); 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":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-1d7f8f614531604f","e-2c2a80fe9ff9b06e"],"text":"A110441 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A110441 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}