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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"0ca5a31b56171c82ae22bd594c2dce2ff751acf3167838bcb9d47db202728bbb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3255c4dcfe77aa0b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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,            4;\n  2:          12,           28,           16;\n  3:          84,          244,          224,           64;\n  4:        1008,         3264,         3664,         1664,          256;\n  5:       19152,        66048,        82672,        46272,        11520,        1024;\n  6:      536256,      1925952,      2579008,      1626304,       507648,       74752,       4096;\n  7:    20913984,     77257152,    108285120,     73741888,     26303488,     4945920,     458752,     16384;\n  8:  1087527168,   4101027840,   5939854848,   4267718656,   1662748928,   362401792,   43638784,   2686976,    65536;\n  9: 72864320256, 279118973952, 414374386176, 309696569344, 128475052800, 30931915776, 4373405696, 354582528, 15138816, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+4+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^2 - 2*n + 4 + 4*x)*P[n - 1, 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] = (n^2 - 2*n + 4 + 4*x)*v[i - 1]); 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-54284d24a094e2d3","steps":["basis A013609","basis A013609"],"edges":["e-327e7f94e2e44f03","e-762cb3f054e5cf71"],"text":"A269944 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A269944 \u2192 basis A013609 \u2192 basis A013609","chain":null}