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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":"dcec3d1126d1e7d3a102ba64fc9fbee0aa84409c2c898cc6b508bab463bf7935","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-f8135e42b21d6fe3","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/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;\n  2:   3,   12;\n  3:   5,   24;\n  4:  11,   84,   108;\n  5:  21,  192,   324;\n  6:  43,  492,  1296,   864;\n  7:  85, 1128,  3672,  3456;\n  8: 171, 2628, 10692, 15552,  6480;\n  9: 341, 5904, 28188, 51840, 32400;\n","recurrence_sentence":"The row polynomials satisfy (6*n+(9*n^2+13*n)*x+(6*n^3+6*n^2)*x^2)*P(n,x) = (30*n-24+(63*n^2+119*n-160)*x+(30*n^3+222*n^2+72*n-312)*x^2+(72*n^3+144*n^2-72*n-144)*x^3)*P(n-1,x) - (9+(27*n+90)*x+(18*n^2+207*n+329)*x^2+(102*n^2+522*n+516)*x^3+(144*n^2+432*n+288)*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[((30*n - 24 + (63*n^2 + 119*n - 160)*x + (30*n^3 + 222*n^2 + 72*n - 312)*x^2 + (72*n^3 + 144*n^2 - 72*n - 144)*x^3)*P[n - 1, x] - (9 + (27*n + 90)*x + (18*n^2 + 207*n + 329)*x^2 + (102*n^2 + 522*n + 516)*x^3 + (144*n^2 + 432*n + 288)*x^4)*D[P[n - 1, x], x])/(6*n + (9*n^2 + 13*n)*x + (6*n^3 + 6*n^2)*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] = ((30*n - 24 + (63*n^2 + 119*n - 160)*x + (30*n^3 + 222*n^2 + 72*n - 312)*x^2 + (72*n^3 + 144*n^2 - 72*n - 144)*x^3)*v[i - 1] - (9 + (27*n + 90)*x + (18*n^2 + 207*n + 329)*x^2 + (102*n^2 + 522*n + 516)*x^3 + (144*n^2 + 432*n + 288)*x^4)*deriv(v[i - 1], x))/(6*n + (9*n^2 + 13*n)*x + (6*n^3 + 6*n^2)*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-e3d901814302c059","steps":["basis A013610","\u03b8+1"],"edges":["e-114dd09068d0c756","e-7ac125cbcff28b8f"],"text":"A128099 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A128099 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}