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n.","chain":null}},{"id":"l-f06351a2e4ab91e0","kind":"fingerprint_related","node_a_id":"n-c9bb7a76ea8cdc0d","node_b_id":"n-6ba8bf97eb6bdb79","fingerprint":"f6-a83eeccfb5efd735","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a83eeccfb5efd735"},"other_id":"n-c9bb7a76ea8cdc0d","other":{"id":"n-c9bb7a76ea8cdc0d","name":"mobius_x_over_x_plus_one of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-a83eeccfb5efd735","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","1","6","6"]],"row_sums_preview":["1","1","3","13","75","541","4683","47293"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["reverse (shift +1)","x \u21a6 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(n+(n-1)*x)*P(n-1,x) - (x+x^2)*d/dx 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 + (n - 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], 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 + (n - 1)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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-efd1fdf07e5468d8","steps":["reverse (shift +1)","reverse","x \u21a6 x+1"],"edges":["e-61bc78adfe1eb25a","e-52ca2f2e23126004","e-32cceb5c6cccf39f"],"text":"A173018 \u2192 reverse (shift +1) \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A173018 \u2192 reverse (shift +1) \u2192 reverse \u2192 x \u21a6 x+1","chain":null}