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begins:\n  0:   1;\n  1:   4,   4;\n  2:   9,  18,    9;\n  3:  16,  48,   48,   16;\n  4:  25, 100,  150,  100,    25;\n  5:  36, 180,  360,  360,   180,    36;\n  6:  49, 294,  735,  980,   735,   294,   49;\n  7:  64, 448, 1344, 2240,  2240,  1344,  448,   64;\n  8:  81, 648, 2268, 4536,  5670,  4536, 2268,  648,  81;\n  9: 100, 900, 3600, 8400, 12600, 12600, 8400, 3600, 900, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n-8+(4*n-8)*x)*P(n-1,x) + (4*n+1+(8*n+2)*x+(4*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x)*P[n - 1, x] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n - 8 + (4*n - 8)*x)*v[i - 1] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*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":"A135065","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"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:   9,  18,    9;\n  3:  16,  48,   48,   16;\n  4:  25, 100,  150,  100,    25;\n  5:  36, 180,  360,  360,   180,    36;\n  6:  49, 294,  735,  980,   735,   294,   49;\n  7:  64, 448, 1344, 2240,  2240,  1344,  448,   64;\n  8:  81, 648, 2268, 4536,  5670,  4536, 2268,  648,  81;\n  9: 100, 900, 3600, 8400, 12600, 12600, 8400, 3600, 900, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n-8+(4*n-8)*x)*P(n-1,x) + (4*n+1+(8*n+2)*x+(4*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x)*P[n - 1, x] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n - 8 + (4*n - 8)*x)*v[i - 1] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*v[i - 2]); 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":"A135065","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:   9,  18,    9;\n  3:  16,  48,   48,   16;\n  4:  25, 100,  150,  100,    25;\n  5:  36, 180,  360,  360,   180,    36;\n  6:  49, 294,  735,  980,   735,   294,   49;\n  7:  64, 448, 1344, 2240,  2240,  1344,  448,   64;\n  8:  81, 648, 2268, 4536,  5670,  4536, 2268,  648,  81;\n  9: 100, 900, 3600, 8400, 12600, 12600, 8400, 3600, 900, 100;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(4*n-8+(4*n-8)*x)*P(n-1,x) + (4*n+1+(8*n+2)*x+(4*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n - 8 + (4*n - 8)*x)*P[n - 1, x] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*P[n - 2, 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; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n - 8 + (4*n - 8)*x)*v[i - 1] + (4*n + 1 + (8*n + 2)*x + (4*n + 1)*x^2)*v[i - 2]); 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-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)"],"edges":["e-b78deffc64bc1942"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2)"},"headline":"A135065: A127733 * A007318 as infinite lower triangular matrices.","chain":"A007318 \u2192 \u00d7(1+2n+n\u00b2)"}