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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":"fba5a9e5294d0a77925957d9f8a3662afa490b8f326727b598c70ab26d5c8f4b","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-2422ccdf8fc66e9a","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","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: 6, 48, 36;\n 2: 36, 576, 1728, 1536, 400;\n 3: 216, 5184, 27216, 55296, 50400, 20736, 3136;\n 4: 1296, 41472, 311040, 995328, 1641600, 1492992, 752640, 196608, 20736;\n 5: 7776, 311040, 3032640, 13271040, 31968000, 46282752, 41771520, 23592960, 8087040, 1536000, 123904;\n 6: 46656, 2239488, 26873856, 149299200, 474336000, 949542912, 1257209856, 1125384192, 683043840, 276480000, 71368704, 10616832, 692224;\n 7: 279936, 15676416, 223388928, 1504935936, 5943974400, 15237476352, 26771254272, 33209450496, 29503014912, 18811699200, 8524099584, 2675441664, 552394752, 67436544, 3686400;\n 8: 1679616, 107495424, 1773674496, 14046068736, 66363494400, 207681159168, 456107065344, 727210524672, 858404708352, 757510963200, 500580089856, 246140633088, 88615747584, 22658678784, 3892838400, 402653184, 18939904;\n 9: 10077696, 725594112, 13604889600, 123834728448, 681252249600, 2519262756864, 6649057787904, 12988887072768, 19210829709312, 21807238348800, 19131772796928, 12988887072768, 6794804330496, 2709330591744, 807410073600, 173946175488, 25568870400, 2293235712, 94633984;\n","recurrence_sentence":"The row polynomials satisfy (9+(54*n-18)*x+(36*n^2+48*n-36)*x^2+(48*n^2-12*n-12)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (54+(324*n+324)*x+(216*n^2+1368*n+756)*x^2+(720*n^2+2160*n+864)*x^3+(816*n^2+1584*n+504)*x^4+(384*n^2+528*n+144)*x^5+(64*n^2+64*n+16)*x^6)*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[((54 + (324*n + 324)*x + (216*n^2 + 1368*n + 756)*x^2 + (720*n^2 + 2160*n + 864)*x^3 + (816*n^2 + 1584*n + 504)*x^4 + (384*n^2 + 528*n + 144)*x^5 + (64*n^2 + 64*n + 16)*x^6)*P[n - 1, x])/(9 + (54*n - 18)*x + (36*n^2 + 48*n - 36)*x^2 + (48*n^2 - 12*n - 12)*x^3 + (16*n^2 - 16*n + 4)*x^4)]];\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] = ((54 + (324*n + 324)*x + (216*n^2 + 1368*n + 756)*x^2 + (720*n^2 + 2160*n + 864)*x^3 + (816*n^2 + 1584*n + 504)*x^4 + (384*n^2 + 528*n + 144)*x^5 + (64*n^2 + 64*n + 16)*x^6)*v[i - 1])/(9 + (54*n - 18)*x + (36*n^2 + 48*n - 36)*x^2 + (48*n^2 - 12*n - 12)*x^3 + (16*n^2 - 16*n + 4)*x^4)); 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-d3c3090977105807","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-e79851ce4a3e511f","e-2f044215bc50d6a9"],"text":"A272867 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A272867 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}