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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":"dc6773edb3ff8461b6711afc137b6c912d3439269e0d55fafb12e84afbd0290e","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-f7935c36a3c0ec8c","role":"sequence","name":"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: 1,   4;\n   3: 1,  24,     9;\n   4: 1,  60,   135,     16;\n   5: 1, 112,   630,    448,      25;\n   6: 1, 180,  1890,   3360,    1125,      36;\n   7: 1, 264,  4455,  14784,   12375,    2376,     49;\n   8: 1, 364,  9009,  48048,   75075,   36036,   4459,     64;\n   9: 1, 480, 16380, 128128,  321750,  288288,  89180,   7680,    81;\n  10: 1, 612, 27540, 297024, 1093950, 1575288, 909636, 195840, 12393, 100;\n","recurrence_sentence":"The row polynomials satisfy (9*n-18-(22*n^3-111*n^2+182*n-96)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (9*n-18-(22*n^3-111*n^2+74*n+120)*x+(4*n^5-28*n^4-37*n^3+329*n^2-357*n+14)*x^2+(12*n^5-52*n^4+63*n^3-30*n+4)*x^3)*P(n-1,x) + (18*x-(44*n^2-68*n+14)*x^2+(8*n^4-16*n^3+42*n^2-54*n-6)*x^3-(8*n^4-16*n^3-2*n^2+14*n-2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n - 18 - (22*n^3 - 111*n^2 + 74*n + 120)*x + (4*n^5 - 28*n^4 - 37*n^3 + 329*n^2 - 357*n + 14)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*P[n - 1, x] + (18*x - (44*n^2 - 68*n + 14)*x^2 + (8*n^4 - 16*n^3 + 42*n^2 - 54*n - 6)*x^3 - (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*D[P[n - 1, x], x])/(9*n - 18 - (22*n^3 - 111*n^2 + 182*n - 96)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((9*n - 18 - (22*n^3 - 111*n^2 + 74*n + 120)*x + (4*n^5 - 28*n^4 - 37*n^3 + 329*n^2 - 357*n + 14)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*v[i - 1] + (18*x - (44*n^2 - 68*n + 14)*x^2 + (8*n^4 - 16*n^3 + 42*n^2 - 54*n - 6)*x^3 - (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*deriv(v[i - 1], x))/(9*n - 18 - (22*n^3 - 111*n^2 + 182*n - 96)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*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-f7935c36a3c0ec8c","steps":["(\u03b8+1)\u00b2"],"edges":["e-1b0ecf2b3e1dc4df"],"text":"A086645 \u2192 (\u03b8+1)\u00b2"},"headline":"A086645 \u2192 (\u03b8+1)\u00b2","chain":null}