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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4 + (3*n + 1)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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}},{"oeis_id":"A257622","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      4,         4;\n  2:     16,        56,         16;\n  3:     64,       552,        552,          64;\n  4:    256,      4696,      11040,        4696,         256;\n  5:   1024,     36968,     171448,      171448,       36968,        1024;\n  6:   4096,    278232,    2305968,     4457648,     2305968,      278232,        4096;\n  7:  16384,   2037736,   28346088,    94844912,    94844912,    28346088,     2037736,      16384;\n  8:  65536,  14673752,  328291072,  1771559528,  3035037184,  1771559528,   328291072,   14673752,     65536;\n  9: 262144, 104551272, 3649754520, 30252677448, 82220225976, 82220225976, 30252677448, 3649754520, 104551272, 262144;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+(3*n+1)*x)*P(n-1,x) + (3*x-3*x^2)*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[(4 + (3*n + 1)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (4 + (3*n + 1)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-e224ad3e3147ef52","steps":[],"edges":[],"text":"A257622"},"headline":"A257622: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.","chain":null}
