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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":"95b6bc1a0d01db2e640328eef036cc437ef7349b17e2dbb42b27da9d560caf45","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-273f45fc84d8c52d","role":"sequence","name":"basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  1,    4;\n   3:  2,   12,    12;\n   4:  3,   28,    60,    32;\n   5:  5,   60,   192,   224,     80;\n   6:  8,  120,   516,   928,    720,    192;\n   7: 13,  232,  1248,  3040,   3680,   2112,    448;\n   8: 21,  436,  2820,  8672,  14320,  12864,   5824,   1024;\n   9: 34,  804,  6072, 22560,  47280,  58176,  41216,  15360,  2304;\n  10: 55, 1460, 12612, 54944, 139680, 218880, 212800, 123904, 39168, 5120;\n","recurrence_sentence":"The row polynomials satisfy (20*n-20+(20*n^2+12*n-32)*x+(8*n^3+8*n^2-16*n)*x^2)*P(n,x) = -(40*n-100+(60*n^2-116*n-160)*x+(16*n^3-8*n^2-200*n-48)*x^2-(32*n^2+64*n)*x^3)*P(n-1,x) + (25+(50*n+90)*x+(20*n^2+160*n+92)*x^2+(48*n^2+136*n+24)*x^3+(16*n^2+32*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(40*n - 100 + (60*n^2 - 116*n - 160)*x + (16*n^3 - 8*n^2 - 200*n - 48)*x^2 - (32*n^2 + 64*n)*x^3)*P[n - 1, x] + (25 + (50*n + 90)*x + (20*n^2 + 160*n + 92)*x^2 + (48*n^2 + 136*n + 24)*x^3 + (16*n^2 + 32*n)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 + 12*n - 32)*x + (8*n^3 + 8*n^2 - 16*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(40*n - 100 + (60*n^2 - 116*n - 160)*x + (16*n^3 - 8*n^2 - 200*n - 48)*x^2 - (32*n^2 + 64*n)*x^3)*v[i - 1] + (25 + (50*n + 90)*x + (20*n^2 + 160*n + 92)*x^2 + (48*n^2 + 136*n + 24)*x^3 + (16*n^2 + 32*n)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 + 12*n - 32)*x + (8*n^3 + 8*n^2 - 16*n)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":["basis A013609","\u03b8+1"],"edges":["e-6b05245c845c9a6d","e-9890eca019d7ca76"],"text":"A026729 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A026729 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}