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Expand[(1 + 2*x + (2*n - 1)*x^2)*P[n - 1, x] + (x - x^3)*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] = (1 + 2*x + (2*n - 1)*x^2)*v[i - 1] + (x - x^3)*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}},{"oeis_id":"A165891","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,    2,     1;\n  2: 1,    6,    10,      6,       1;\n  3: 1,   14,    47,     68,      47,       14,         1;\n  4: 1,   30,   176,    450,     606,      450,       176,        30,         1;\n  5: 1,   62,   597,   2392,    5162,     6612,      5162,      2392,       597,        62,         1;\n  6: 1,  126,  1926,  11382,   35967,    69132,     85492,     69132,     35967,     11382,      1926,       126,         1;\n  7: 1,  254,  6043,  50892,  223785,   600546,   1060411,   1277096,   1060411,    600546,    223785,     50892,      6043,      254,       1;\n  8: 1,  510, 18652, 219210, 1299268,  4661550,  11085604,  18343050,  21641590,  18343050,  11085604,   4661550,   1299268,   219210,   18652,    510,     1;\n  9: 1, 1022, 56993, 922304, 7214540, 33636776, 103812812, 224854208, 353402054, 410144180, 353402054, 224854208, 103812812, 33636776, 7214540, 922304, 56993, 1022, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x+(2*n-1)*x^2)*P(n-1,x) + (x-x^3)*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[(1 + 2*x + (2*n - 1)*x^2)*P[n - 1, x] + (x - x^3)*D[P[n - 1, x], x]];\nTable[Reverse[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] = (1 + 2*x + (2*n - 1)*x^2)*v[i - 1] + (x - x^3)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = Vecrev(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":true,"trim_zeros":"none","regular_triangle":false}}],"derivation":null,"headline":"A165891: Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","chain":null}