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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A152842","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3;\n  2: 1,  4,   3;\n  3: 1,  7,  15,   9;\n  4: 1,  8,  22,  24,    9;\n  5: 1, 11,  46,  90,   81,   27;\n  6: 1, 12,  57, 136,  171,  108,   27;\n  7: 1, 15,  93, 307,  579,  621,  351,   81;\n  8: 1, 16, 108, 400,  886, 1200,  972,  432,   81;\n  9: 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x + 3*x^2)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + 3*x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-7196be931c2ed996","steps":[],"edges":[],"text":"A152842"},"headline":"A152842: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","chain":null}