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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"1f45984ea0f276ee0c71bf7091f6057eb3684aa215b554b0721c0ec414219458","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,               2;\n  2:               2,               6,               4;\n  3:              18,              58,              48,               8;\n  4:             504,            1660,            1460,             320,             16;\n  5:           32760,          108908,           98220,           23720,           1680,           32;\n  6:         4127760,        13787928,        12593536,         3185160,         259120,         7392,         64;\n  7:       895723920,      3000235896,      2760373168,       716366792,       62599360,      2122304,      28672,      128;\n  8:    308129028480,   1033872596064,    955568841584,    251950922784,    22966913424,    855271296,   14107776,   101376,    256;\n  9: 158070191610240, 530992899837792, 492274560924720, 131161961071360, 12285928432080, 484688001696, 8947831680, 80221440, 334080, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+2*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 + 2*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 + 2*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 A013609"],"edges":["e-e62c7427c0d251dd"],"text":"A269947 \u2192 basis A013609"},"headline":"A269947 \u2192 basis A013609","chain":null}