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v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (3*n - 2)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x) + 3*x*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":"A144440","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: 1,      1;\n   3: 1,     11,        1;\n   4: 1,     54,       54,         1;\n   5: 1,    229,      789,       229,          1;\n   6: 1,    932,     7975,      7975,        932,          1;\n   7: 1,   3747,    68628,    161867,      68628,       3747,         1;\n   8: 1,  15010,   543144,   2534759,    2534759,     543144,     15010,        1;\n   9: 1,  60065,  4098439,  34243778,   66389335,   34243778,   4098439,    60065,      1;\n  10: 1, 240288, 30055533, 421937553, 1418566080, 1418566080, 421937553, 30055533, 240288, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) + 3*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n - 5)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x] + 3*x*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] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (3*n - 5)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x) + 3*x*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}},{"oeis_id":"A144440","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      1;\n   3: 1,     11,        1;\n   4: 1,     54,       54,         1;\n   5: 1,    229,      789,       229,          1;\n   6: 1,    932,     7975,      7975,        932,          1;\n   7: 1,   3747,    68628,    161867,      68628,       3747,         1;\n   8: 1,  15010,   543144,   2534759,    2534759,     543144,     15010,        1;\n   9: 1,  60065,  4098439,  34243778,   66389335,   34243778,   4098439,    60065,      1;\n  10: 1, 240288, 30055533, 421937553, 1418566080, 1418566080, 421937553, 30055533, 240288, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) + 3*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (3*n - 5)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x] + 3*x*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (3*n - 5)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x) + 3*x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-ef0a5df5e949a196","steps":[],"edges":[],"text":"A144440"},"headline":"A144440: Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","chain":null}