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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":"54220a82213bdfe442d24dd9e5177c94340ac3c62ca0a3df91b3b31952aab345","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-8e6b51ac707373f6","role":"sequence","name":"Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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:      1,       2;\n  2:      3,      10,        6;\n  3:     12,      56,       63,       20;\n  4:     55,     330,      540,      336,       70;\n  5:    273,    2002,     4290,     3960,     1650,      252;\n  6:   1428,   12376,    32760,    40040,    25025,     7722,      924;\n  7:   7752,   77520,   244188,   371280,   309400,   144144,    35035,    3432;\n  8:  43263,  490314,  1790712,  3255840,  3391500,  2116296,   779688,  155584,  12870;\n  9: 246675, 3124550, 12978900, 27457584, 34321980, 26860680, 13294680, 4031040, 680238, 48620;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3+2*n^2-2*n+(8*n^4+4*n^3)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(54*n^4+90*n^3-174*n^2+76*n-10)*x+(48*n^4+15*n^3-66*n^2+31*n-4)*x^2+(2*n^4-2*n^3-2*n^2+2*n)*x^3)*P(n-1,x) - ((27*n^2-18*n+3)*x+(18*n^3+48*n^2-35*n+5)*x^2+(20*n^3+21*n^2-19*n+2)*x^3+(2*n^3-2*n)*x^4)*d/dx 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[((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 90*n^3 - 174*n^2 + 76*n - 10)*x + (48*n^4 + 15*n^3 - 66*n^2 + 31*n - 4)*x^2 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^3)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 48*n^2 - 35*n + 5)*x^2 + (20*n^3 + 21*n^2 - 19*n + 2)*x^3 + (2*n^3 - 2*n)*x^4)*D[P[n - 1, x], x])/(12*n^3 + 2*n^2 - 2*n + (8*n^4 + 4*n^3)*x)]];\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] = ((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 90*n^3 - 174*n^2 + 76*n - 10)*x + (48*n^4 + 15*n^3 - 66*n^2 + 31*n - 4)*x^2 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^3)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 48*n^2 - 35*n + 5)*x^2 + (20*n^3 + 21*n^2 - 19*n + 2)*x^3 + (2*n^3 - 2*n)*x^4)*deriv(v[i - 1], x))/(12*n^3 + 2*n^2 - 2*n + (8*n^4 + 4*n^3)*x)); 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-8e6b51ac707373f6","steps":["\u03b8+1"],"edges":["e-26f78ea6765c58ae"],"text":"A108426 \u2192 \u03b8+1"},"headline":"A108426 \u2192 \u03b8+1","chain":null}