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begins:\n  0: 1;\n  1: 1,        1;\n  2: 1,       18,           1;\n  3: 1,      179,         179,            1;\n  4: 1,     1636,        6086,         1636,            1;\n  5: 1,    14757,      144362,       144362,        14757,            1;\n  6: 1,   132854,     2941135,      7218100,      2941135,       132854,            1;\n  7: 1,  1195735,    55446309,    277509955,    277509955,     55446309,      1195735,           1;\n  8: 1, 10761672,  1001178268,   9211047544,  18315657030,   9211047544,   1001178268,    10761672,        1;\n  9: 1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(8*n-7)*x)*P(n-1,x) + (8*x-8*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[(1 + (8*n - 7)*x)*P[n - 1, x] + (8*x - 8*x^2)*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 + (8*n - 7)*x)*v[i - 1] + (8*x - 8*x^2)*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":"A167884","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,       18,           1;\n   4: 1,      179,         179,            1;\n   5: 1,     1636,        6086,         1636,            1;\n   6: 1,    14757,      144362,       144362,        14757,            1;\n   7: 1,   132854,     2941135,      7218100,      2941135,       132854,            1;\n   8: 1,  1195735,    55446309,    277509955,    277509955,     55446309,      1195735,           1;\n   9: 1, 10761672,  1001178268,   9211047544,  18315657030,   9211047544,   1001178268,    10761672,        1;\n  10: 1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(8*n-15)*x)*P(n-1,x) + (8*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (8*n - 15)*x)*P[n - 1, x] + (8*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (8*n - 15)*x)*v[i - 1] + (8*x - 8*x^2)*deriv(v[i - 1], x)); 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":"A167884","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,       18,           1;\n   4: 1,      179,         179,            1;\n   5: 1,     1636,        6086,         1636,            1;\n   6: 1,    14757,      144362,       144362,        14757,            1;\n   7: 1,   132854,     2941135,      7218100,      2941135,       132854,            1;\n   8: 1,  1195735,    55446309,    277509955,    277509955,     55446309,      1195735,           1;\n   9: 1, 10761672,  1001178268,   9211047544,  18315657030,   9211047544,   1001178268,    10761672,        1;\n  10: 1, 96855113, 17633445860, 279333923732, 982069631294, 982069631294, 279333923732, 17633445860, 96855113, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(8*n-15)*x)*P(n-1,x) + (8*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (8*n - 15)*x)*P[n - 1, x] + (8*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (8*n - 15)*x)*v[i - 1] + (8*x - 8*x^2)*deriv(v[i - 1], x)); 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-90af30f849fc554d","steps":[],"edges":[],"text":"A167884"},"headline":"A167884: Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","chain":null}