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v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1 + (2*n + 1)*x)*v[i - 1] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*v[i - 2])/(n + 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":"A001263","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,  3,   1;\n   4: 1,  6,   6,    1;\n   5: 1, 10,  20,   10,    1;\n   6: 1, 15,  50,   50,   15,    1;\n   7: 1, 21, 105,  175,  105,   21,    1;\n   8: 1, 28, 196,  490,  490,  196,   28,   1;\n   9: 1, 36, 336, 1176, 1764, 1176,  336,  36,  1;\n  10: 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-2-(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n + 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n + 1)); 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":"A001263","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,  3,   1;\n   4: 1,  6,   6,    1;\n   5: 1, 10,  20,   10,    1;\n   6: 1, 15,  50,   50,   15,    1;\n   7: 1, 21, 105,  175,  105,   21,    1;\n   8: 1, 28, 196,  490,  490,  196,   28,   1;\n   9: 1, 36, 336, 1176, 1764, 1176,  336,  36,  1;\n  10: 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-2-(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n + 1)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n + 1)); 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-c2eaf5212a4dec13","steps":[],"edges":[],"text":"A001263"},"headline":"A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","chain":null}