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begins:\n   1: 1;\n   2: 0, 1;\n   3: 0, 1, 1;\n   4: 0, 0, 2, 1;\n   5: 0, 0, 1, 3, 1;\n   6: 0, 0, 0, 3, 4,  1;\n   7: 0, 0, 0, 1, 6,  5,  1;\n   8: 0, 0, 0, 0, 4, 10,  6,  1;\n   9: 0, 0, 0, 0, 1, 10, 15,  7, 1;\n  10: 0, 0, 0, 0, 0,  5, 20, 21, 8, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = x*v[i - 1] + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A011973","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1, 1;\n  3: 1, 2;\n  4: 1, 3,  1;\n  5: 1, 4,  3;\n  6: 1, 5,  6,  1;\n  7: 1, 6, 10,  4;\n  8: 1, 7, 15, 10, 1;\n  9: 1, 8, 21, 20, 5;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false}},{"oeis_id":"A026729","target":"triangle","first_row":1,"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: 0, 1;\n  2: 0, 1, 1;\n  3: 0, 0, 2, 1;\n  4: 0, 0, 1, 3, 1;\n  5: 0, 0, 0, 3, 4,  1;\n  6: 0, 0, 0, 1, 6,  5,  1;\n  7: 0, 0, 0, 0, 4, 10,  6,  1;\n  8: 0, 0, 0, 0, 1, 10, 15,  7, 1;\n  9: 0, 0, 0, 0, 0,  5, 20, 21, 8, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*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] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A052553","target":"reversed_triangle","first_row":1,"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, 0;\n  2: 1, 1,  0;\n  3: 1, 2,  0,  0;\n  4: 1, 3,  1,  0, 0;\n  5: 1, 4,  3,  0, 0, 0;\n  6: 1, 5,  6,  1, 0, 0, 0;\n  7: 1, 6, 10,  4, 0, 0, 0, 0;\n  8: 1, 7, 15, 10, 1, 0, 0, 0, 0;\n  9: 1, 8, 21, 20, 5, 0, 0, 0, 0, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*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] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*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":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A098925","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1,  1;\n  3: 2,  1;\n  4: 1,  3,  1;\n  5: 3,  4,  1;\n  6: 1,  6,  5, 1;\n  7: 4, 10,  6, 1;\n  8: 1, 10, 15, 7, 1;\n  9: 5, 20, 21, 8, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];\nTable[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":[],"edges":[],"text":"A026729"},"headline":"A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","chain":null}