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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A091042","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: 1,   3;\n  2: 1,  10,    5;\n  3: 1,  21,   35,     7;\n  4: 1,  36,  126,    84,     9;\n  5: 1,  55,  330,   462,   165,    11;\n  6: 1,  78,  715,  1716,  1287,   286,    13;\n  7: 1, 105, 1365,  5005,  6435,  3003,   455,    15;\n  8: 1, 136, 2380, 12376, 24310, 19448,  6188,   680,  17;\n  9: 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 19;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + 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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + 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}},{"oeis_id":"A103327","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:  3,   1;\n  2:  5,  10,     1;\n  3:  7,  35,    21,     1;\n  4:  9,  84,   126,    36,     1;\n  5: 11, 165,   462,   330,    55,     1;\n  6: 13, 286,  1287,  1716,   715,    78,     1;\n  7: 15, 455,  3003,  6435,  5005,  1365,   105,    1;\n  8: 17, 680,  6188, 19448, 24310, 12376,  2380,  136,   1;\n  9: 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + 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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + 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}}],"derivation":{"root":"n-c30cb06ad75c5e21","steps":[],"edges":[],"text":"A103327"},"headline":"A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","chain":null}