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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":"a80f1d6c43d5e33ddf73b620e41c5122e622a5f81ab239a4adcb826fdf54eb72","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-563143c7a3f59683","role":"sequence","name":"shift_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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:      2,       2;\n  2:      6,      12,       3;\n  3:     22,      66,      39,        4;\n  4:     92,     368,     348,       96,       5;\n  5:    422,    2110,    2760,     1300,     200,       6;\n  6:   2074,   12444,   20835,    14080,    3925,     372,      7;\n  7:  10754,   75278,  153783,   136220,   56455,   10206,    637,     8;\n  8:  58202,  465616, 1123500,  1234688,  686630,  190848,  23716,  1024,    9;\n  9: 326240, 2936160, 8174340, 10742760, 7535850, 2875320, 566538, 50448, 1566, 10;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2+84*n+144+(6*n^3+54*n^2+156*n+144)*x+(3*n^4+30*n^3+105*n^2+150*n+72)*x^2+(n^5+10*n^4+35*n^3+50*n^2+24*n)*x^3)*P(n,x) = (132*n^2+204*n+144+(114*n^3+474*n^2+468*n+144)*x+(45*n^4+306*n^3+603*n^2+414*n+72)*x^2+(11*n^5+110*n^4+325*n^3+370*n^2+144*n)*x^3+(10*n^5+55*n^4+98*n^3+65*n^2+12*n)*x^4)*P(n-1,x) - (72*n-144+(108*n^2+252*n-288)*x+(60*n^3+348*n^2+324*n-252)*x^2+(12*n^4+153*n^3+402*n^2+180*n-144)*x^3+(27*n^4+162*n^3+249*n^2+60*n-36)*x^4+(15*n^4+69*n^3+87*n^2+24*n)*x^5)*d/dx P(n-1,x) + (48+(48*n+192)*x+(36*n^2+228*n+348)*x^2+(12*n^3+146*n^2+440*n+372)*x^3+(30*n^3+212*n^2+425*n+240)*x^4+(24*n^3+130*n^2+202*n+84)*x^5+(6*n^3+28*n^2+37*n+12)*x^6)*d^2/dx^2 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[((132*n^2 + 204*n + 144 + (114*n^3 + 474*n^2 + 468*n + 144)*x + (45*n^4 + 306*n^3 + 603*n^2 + 414*n + 72)*x^2 + (11*n^5 + 110*n^4 + 325*n^3 + 370*n^2 + 144*n)*x^3 + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x^4)*P[n - 1, x] - (72*n - 144 + (108*n^2 + 252*n - 288)*x + (60*n^3 + 348*n^2 + 324*n - 252)*x^2 + (12*n^4 + 153*n^3 + 402*n^2 + 180*n - 144)*x^3 + (27*n^4 + 162*n^3 + 249*n^2 + 60*n - 36)*x^4 + (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^5)*D[P[n - 1, x], x] + (48 + (48*n + 192)*x + (36*n^2 + 228*n + 348)*x^2 + (12*n^3 + 146*n^2 + 440*n + 372)*x^3 + (30*n^3 + 212*n^2 + 425*n + 240)*x^4 + (24*n^3 + 130*n^2 + 202*n + 84)*x^5 + (6*n^3 + 28*n^2 + 37*n + 12)*x^6)*D[P[n - 1, x], {x, 2}])/(12*n^2 + 84*n + 144 + (6*n^3 + 54*n^2 + 156*n + 144)*x + (3*n^4 + 30*n^3 + 105*n^2 + 150*n + 72)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*x^3)]];\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] = ((132*n^2 + 204*n + 144 + (114*n^3 + 474*n^2 + 468*n + 144)*x + (45*n^4 + 306*n^3 + 603*n^2 + 414*n + 72)*x^2 + (11*n^5 + 110*n^4 + 325*n^3 + 370*n^2 + 144*n)*x^3 + (10*n^5 + 55*n^4 + 98*n^3 + 65*n^2 + 12*n)*x^4)*v[i - 1] - (72*n - 144 + (108*n^2 + 252*n - 288)*x + (60*n^3 + 348*n^2 + 324*n - 252)*x^2 + (12*n^4 + 153*n^3 + 402*n^2 + 180*n - 144)*x^3 + (27*n^4 + 162*n^3 + 249*n^2 + 60*n - 36)*x^4 + (15*n^4 + 69*n^3 + 87*n^2 + 24*n)*x^5)*deriv(v[i - 1], x) + (48 + (48*n + 192)*x + (36*n^2 + 228*n + 348)*x^2 + (12*n^3 + 146*n^2 + 440*n + 372)*x^3 + (30*n^3 + 212*n^2 + 425*n + 240)*x^4 + (24*n^3 + 130*n^2 + 202*n + 84)*x^5 + (6*n^3 + 28*n^2 + 37*n + 12)*x^6)*deriv(deriv(v[i - 1], x), x))/(12*n^2 + 84*n + 144 + (6*n^3 + 54*n^2 + 156*n + 144)*x + (3*n^4 + 30*n^3 + 105*n^2 + 150*n + 72)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*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-047228a38014e6d4","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-0c759266bd791ca5","e-a611c4892070c85c"],"text":"A056939 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A056939 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}