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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"3b4a0e900bd154e696cea265849dd57d3add290258c33b59394c1256c05c8357","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-b12c32f9a9e37f0c","role":"sequence","name":"Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.","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:     1;\n  3:     7,     24;\n  4:    25,     96;\n  5:    61,    240;\n  6:   481,   3360,    5760;\n  7:  2731,  21000,   40320;\n  8: 10417,  81984,  161280;\n  9: 91225, 969696, 3386880, 3870720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n^2-9*n+6+(12*n^2-36*n+24)*x)*P(n-3,x) + P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n^2 - 9*n + 6 + (12*n^2 - 36*n + 24)*x)*P[n - 3, 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; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (3*n^2 - 9*n + 6 + (12*n^2 - 36*n + 24)*x)*v[i - 3] + 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-b12c32f9a9e37f0c","steps":["basis A013611"],"edges":["e-270d3738022f1c68"],"text":"A118394 \u2192 basis A013611"},"headline":"A118394 \u2192 basis A013611","chain":null}