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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":"8d7ed01d7287fdf7e69a2df605e850a6cabd1e9163b6ddebdf4f20f22c1df2b3","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-914a474e9f3b0e2a","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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:       3,        6;\n   3:      13,       48,        27;\n   4:      63,      342,       405,       108;\n   5:     321,     2304,      4212,      2592,       405;\n   6:    1683,    15030,     37260,     36720,     14175,      1458;\n   7:    8989,    96048,    300915,    410400,    261225,     69984,      5103;\n   8:   48639,   605094,   2292381,   4003020,   3586275,   1622754,    321489,    17496;\n   9:  265729,  3772416,  16768080,  35731584,  41447700,  26780544,   9144576,  1399680,   59049;\n  10: 1462563, 23331942, 119038896, 299630016, 427529340, 361169928, 178319232, 47869056, 5845851, 196830;\n","recurrence_sentence":"The row polynomials satisfy (12*n-12+(36*n^2-63*n+27)*x+(27*n^3-81*n^2+81*n-27)*x^2)*P(n,x) = (84*n-132+(288*n^2-513*n+81)*x+(189*n^3-27*n^2-783*n+513)*x^2+(243*n^3-567*n^2+162*n+162)*x^3)*P(n-1,x) - (8+48*n*x+(36*n^2+198*n-144)*x^2+(162*n^2+135*n-243)*x^3+(162*n^2-81*n-81)*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[((84*n - 132 + (288*n^2 - 513*n + 81)*x + (189*n^3 - 27*n^2 - 783*n + 513)*x^2 + (243*n^3 - 567*n^2 + 162*n + 162)*x^3)*P[n - 1, x] - (8 + 48*n*x + (36*n^2 + 198*n - 144)*x^2 + (162*n^2 + 135*n - 243)*x^3 + (162*n^2 - 81*n - 81)*x^4)*D[P[n - 1, x], x])/(12*n - 12 + (36*n^2 - 63*n + 27)*x + (27*n^3 - 81*n^2 + 81*n - 27)*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] = ((84*n - 132 + (288*n^2 - 513*n + 81)*x + (189*n^3 - 27*n^2 - 783*n + 513)*x^2 + (243*n^3 - 567*n^2 + 162*n + 162)*x^3)*v[i - 1] - (8 + 48*n*x + (36*n^2 + 198*n - 144)*x^2 + (162*n^2 + 135*n - 243)*x^3 + (162*n^2 - 81*n - 81)*x^4)*deriv(v[i - 1], x))/(12*n - 12 + (36*n^2 - 63*n + 27)*x + (27*n^3 - 81*n^2 + 81*n - 27)*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-78ba05b4a3dd925d","steps":["basis A013610","\u03b8+1"],"edges":["e-35b37e0190870340","e-ddc76f9aa87db4cf"],"text":"A104684 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A104684 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}