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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a8b964d12efba7a2ceb462567ae45bd4a41edb5d2cbbe6f7f3fcdc3d195e6a0e","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-fefc87d068bda822","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 3, 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)^3*T(n-1,k), for n>=0 and 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:               1,               1;\n  2:               2,               3,               1;\n  3:              18,              29,              12,              1;\n  4:             504,             830,             365,             40,            1;\n  5:           32760,           54454,           24555,           2965,          105,           1;\n  6:         4127760,         6893964,         3148384,         398145,        16195,         231,         1;\n  7:       895723920,      1500117948,       690093292,       89545849,      3912460,       66322,       448,      1;\n  8:    308129028480,    516936298032,    238892210396,    31493865348,   1435432089,    26727228,    220434,    792,    1;\n  9: 158070191610240, 265496449918896, 123068640231180, 16395245133920, 767870527005, 15146500053, 139809870, 626730, 1305, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+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^3 - 3*n^2 + 3*n + 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^3 - 3*n^2 + 3*n + 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-fefc87d068bda822","steps":["basis A007318"],"edges":["e-0e122e2df114c72b"],"text":"A269947 \u2192 basis A007318"},"headline":"A269947 \u2192 basis A007318","chain":null}