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d":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0,      2;\n  2: 0,      3,       3;\n  3: 0,      8,      12,       4;\n  4: 0,     30,      55,      30,      5;\n  5: 0,    144,     300,     210,     60,      6;\n  6: 0,    840,    1918,    1575,    595,    105,     7;\n  7: 0,   5760,   14112,   12992,   5880,   1400,   168,    8;\n  8: 0,  45360,  117612,  118188,  60921,  17640,  2898,  252,   9;\n  9: 0, 403200, 1095840, 1181240, 672840, 224490, 45360, 5460, 360, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+2*x)*P(n-1,x) + (n-2-(n-3)*x-x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + 2*x)*P[n - 1, x] + (n - 2 - (n - 3)*x - x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + 2*x)*v[i - 1] + (n - 2 - (n - 3)*x - x^2)*v[i - 2]); 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}},{"oeis_id":"A180013","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0,      2;\n   3: 0,      3,       3;\n   4: 0,      8,      12,       4;\n   5: 0,     30,      55,      30,      5;\n   6: 0,    144,     300,     210,     60,      6;\n   7: 0,    840,    1918,    1575,    595,    105,     7;\n   8: 0,   5760,   14112,   12992,   5880,   1400,   168,    8;\n   9: 0,  45360,  117612,  118188,  60921,  17640,  2898,  252,   9;\n  10: 0, 403200, 1095840, 1181240, 672840, 224490, 45360, 5460, 360, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-2+2*x)*P(n-1,x) + (n-3-(n-4)*x-x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[(n - 2 + 2*x)*P[n - 1, x] + (n - 3 - (n - 4)*x - x^2)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = (n - 2 + 2*x)*v[i - 1] + (n - 3 - (n - 4)*x - x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 2x\u22121"],"edges":["e-3176274160cea224"],"text":"A028338 \u2192 x \u21a6 2x\u22121"},"headline":"A180013: Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.","chain":"A028338 \u2192 x \u21a6 2x\u22121"}