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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":"1ea1f7f27969fa8ad3316890b5a61e43e599e2b67bb3bf44d43e62f12450d93d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-91f141c4f7274be3","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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:      2,       4;\n   3:      8,      32,       12;\n   4:     32,     192,      216,       32;\n   5:    128,    1024,     1920,     1024,       80;\n   6:    512,    5120,    13440,    12800,     4000,      192;\n   7:   2048,   24576,    82944,   114688,    67200,    13824,      448;\n   8:   8192,  114688,   473088,   860160,   752640,   301056,    43904,    1024;\n   9:  32768,  524288,  2555904,  5767168,  6758400,  4128768,  1204224,  131072,   2304;\n  10: 131072, 2359296, 13271040, 35782656, 52715520, 43794432, 19869696, 4423680, 373248, 5120;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-7*n+6+(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = (8*n^2-28*n+24+(8*n^3-12*n^2-32*n+48)*x+(12*n^3-38*n^2+24*n+8)*x^2)*P(n-1,x) - ((4*n-6)*x+(4*n^2+4*n-14)*x^2+(8*n^2-8*n-4)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 28*n + 24 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*P[n - 1, x] - ((4*n - 6)*x + (4*n^2 + 4*n - 14)*x^2 + (8*n^2 - 8*n - 4)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 28*n + 24 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (12*n^3 - 38*n^2 + 24*n + 8)*x^2)*v[i - 1] - ((4*n - 6)*x + (4*n^2 + 4*n - 14)*x^2 + (8*n^2 - 8*n - 4)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); 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-f7935c36a3c0ec8c","steps":["basis A013609","\u03b8+1"],"edges":["e-55a1228129c4451e","e-ae97c847df30fb4d"],"text":"A086645 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A086645 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}