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begins:\n  0: 1;\n  1: 1,  2;\n  2: 1,  4,   2;\n  3: 1,  6,   9,   2;\n  4: 1,  8,  20,  16,    2;\n  5: 1, 10,  35,  50,   25,    2;\n  6: 1, 12,  54, 112,  105,   36,    2;\n  7: 1, 14,  77, 210,  294,  196,   49,   2;\n  8: 1, 16, 104, 352,  660,  672,  336,  64,  2;\n  9: 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x)*P(n-1,x) - (n-2)*x^2*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 4*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n - 4)*x)*P[n - 1, x] - (n - 2)*x^2*P[n - 2, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 4*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (2*n - 4)*x)*v[i - 1] - (n - 2)*x^2*v[i - 2])/(n - 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":"A084534","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,  2;\n  2: 1,  4,   2;\n  3: 1,  6,   9,   2;\n  4: 1,  8,  20,  16,    2;\n  5: 1, 10,  35,  50,   25,    2;\n  6: 1, 12,  54, 112,  105,   36,    2;\n  7: 1, 14,  77, 210,  294,  196,   49,   2;\n  8: 1, 16, 104, 352,  660,  672,  336,  64,  2;\n  9: 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x)*P(n-1,x) - (n-2)*x^2*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 4*x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 4*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n - 4)*x)*P[n - 1, x] - (n - 2)*x^2*P[n - 2, x])/(n - 2)]];\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 + 2*x; v[3] = 1 + 4*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (2*n - 4)*x)*v[i - 1] - (n - 2)*x^2*v[i - 2])/(n - 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-2496e2f555756362","steps":[],"edges":[],"text":"A084534"},"headline":"A084534: Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","chain":null}