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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":"3ebc4f87262e55c9faf628521c970dbf93749eeb086682e0c39007f05e7d042e","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-8409b69bc5d8591b","role":"sequence","name":"basis_map_A013610 of Exponential Riordan array (1, x(1+x/2)).","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,     6;\n  2:    2,    18,      27;\n  3:    4,    54,     162,     108;\n  4:   10,   168,     729,    1080,      405;\n  5:   26,   540,    3105,    7020,     6075,     1458;\n  6:   76,  1836,   12960,   39420,    54675,    30618,     5103;\n  7:  232,  6468,   54432,  207900,   396900,   367416,   142884,    17496;\n  8:  764, 23904,  232092, 1064448,  2594025,  3388392,  2214702,   629856,   59049;\n  9: 2620, 91368, 1007964, 5397840, 15997905, 27096930, 25504794, 12282192, 2657205, 196830;\n","recurrence_sentence":"The row polynomials satisfy (24+(36*n+72)*x+(27*n^2+27*n)*x^2-162*n*x^3)*P(n,x) = -(24*n-48+(54*n^2-54*n-252)*x+(27*n^3-27*n^2-378*n-324)*x^2-(243*n^2+81*n-162)*x^3+(486*n+486)*x^4)*P(n-1,x) + (8+(24*n+48)*x+(18*n^2+90*n+36)*x^2+(54*n^2-54*n-216)*x^3-(324*n+324)*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[(-(24*n - 48 + (54*n^2 - 54*n - 252)*x + (27*n^3 - 27*n^2 - 378*n - 324)*x^2 - (243*n^2 + 81*n - 162)*x^3 + (486*n + 486)*x^4)*P[n - 1, x] + (8 + (24*n + 48)*x + (18*n^2 + 90*n + 36)*x^2 + (54*n^2 - 54*n - 216)*x^3 - (324*n + 324)*x^4)*D[P[n - 1, x], x])/(24 + (36*n + 72)*x + (27*n^2 + 27*n)*x^2 - 162*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] = (-(24*n - 48 + (54*n^2 - 54*n - 252)*x + (27*n^3 - 27*n^2 - 378*n - 324)*x^2 - (243*n^2 + 81*n - 162)*x^3 + (486*n + 486)*x^4)*v[i - 1] + (8 + (24*n + 48)*x + (18*n^2 + 90*n + 36)*x^2 + (54*n^2 - 54*n - 216)*x^3 - (324*n + 324)*x^4)*deriv(v[i - 1], x))/(24 + (36*n + 72)*x + (27*n^2 + 27*n)*x^2 - 162*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-f6268da8d9636534","steps":["basis A013610","\u03b8+1"],"edges":["e-a15a4571c9a83032","e-de9c87e302d604d5"],"text":"A122848 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A122848 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}