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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":"3d649d8466a3d205ada782dde8c996f830373d4099ef18d0c932f58215e455f2","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-146d64c3583c8509","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:      3,        8;\n  2:     11,       56,       48;\n  3:     45,      336,      576,       256;\n  4:    197,     1936,     4992,      4608,      1280;\n  5:    903,    11000,    37920,     53760,     32000,      6144;\n  6:   4279,    62184,   268560,    515840,    480000,    202752,     28672;\n  7:  20793,   351008,  1822464,   4426240,   5644800,   3784704,   1204224,    131072;\n  8: 103049,  1981472, 12020736,  35338240,  57451520,  53329920,  27295744,   6815744,   589824;\n  9: 518859, 11194344, 77709888, 268326912, 531256320, 633249792, 452788224, 184025088, 37158912, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (2*n+4+(8*n^2+24*n+16)*x+(8*n^3+24*n^2+16*n)*x^2)*P(n,x) = (14*n+4+(64*n^2+112*n+16)*x+(56*n^3+264*n^2+208*n)*x^2+(96*n^3+224*n^2+64*n)*x^3)*P(n-1,x) - (1+(8*n+14)*x+(8*n^2+68*n+64)*x^2+(48*n^2+184*n+112)*x^3+(64*n^2+160*n+64)*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[((14*n + 4 + (64*n^2 + 112*n + 16)*x + (56*n^3 + 264*n^2 + 208*n)*x^2 + (96*n^3 + 224*n^2 + 64*n)*x^3)*P[n - 1, x] - (1 + (8*n + 14)*x + (8*n^2 + 68*n + 64)*x^2 + (48*n^2 + 184*n + 112)*x^3 + (64*n^2 + 160*n + 64)*x^4)*D[P[n - 1, x], x])/(2*n + 4 + (8*n^2 + 24*n + 16)*x + (8*n^3 + 24*n^2 + 16*n)*x^2)]];\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] = ((14*n + 4 + (64*n^2 + 112*n + 16)*x + (56*n^3 + 264*n^2 + 208*n)*x^2 + (96*n^3 + 224*n^2 + 64*n)*x^3)*v[i - 1] - (1 + (8*n + 14)*x + (8*n^2 + 68*n + 64)*x^2 + (48*n^2 + 184*n + 112)*x^3 + (64*n^2 + 160*n + 64)*x^4)*deriv(v[i - 1], x))/(2*n + 4 + (8*n^2 + 24*n + 16)*x + (8*n^3 + 24*n^2 + 16*n)*x^2)); 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-8ce61d302cec295b","steps":["basis A013609","\u03b8+1"],"edges":["e-01612a21c966f541","e-b952ffc1e5272221"],"text":"A114687 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A114687 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}