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row polynomials satisfy P(n,x) = (n^2+10*n+25+x)*P(n-1,x) + (2*n+11)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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^2 + 10*n + 25 + x)*P[n - 1, x] + (2*n + 11)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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^2 + 10*n + 25 + x)*v[i - 1] + (2*n + 11)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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":"A371259","target":"triangle","first_row":0,"oeis_offset":3,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   3:                  1;\n   4:                 36,                  1;\n   5:               1764,                100,                 1;\n   6:             112896,               9864,               200,               1;\n   7:            9144576,            1099296,             34064,             344,              1;\n   8:          914457600,          142159392,           6004512,           92200,            540,            1;\n   9:       110649369600,        21385410048,        1156921920,        24075712,         213700,          796,         1;\n  10:     15933509222400,      3724783667712,      248142106368,      6573957120,       78782912,       443744,      1120,       1;\n  11:   2692763058585600,    745991108093952,    59556757600512,   1931075129088,    29342218688,    222555968,    848064,    1520,    1;\n  12: 527781559482777600, 170540762379724800, 15992521053825024, 617637469906944, 11437953984000, 109684923136, 561781568, 1518384, 2004, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+4*n+4+x)*P(n-1,x) + (2*n+5)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 4, with P(3,x) = 1.","mathematica":"P[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^2 + 4*n + 4 + x)*P[n - 1, x] + (2*n + 5)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten","pari":"P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (n^2 + 4*n + 4 + x)*v[i - 1] + (2*n + 5)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};\nT(12)","reading":{"first_row":0,"offset":3,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-b8808e9caf8dcf35","steps":[],"edges":[],"text":"A371259"},"headline":"A371259: Triangle read by rows, (2, 3)-Lah numbers.","chain":null}