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begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  6,    6;\n   4: 1, 12,   30,    20;\n   5: 1, 20,   90,   140,    70;\n   6: 1, 30,  210,   560,   630,    252;\n   7: 1, 42,  420,  1680,  3150,   2772,    924;\n   8: 1, 56,  756,  4200, 11550,  16632,  12012,   3432;\n   9: 1, 72, 1260,  9240, 34650,  72072,  84084,  51480,  12870;\n  10: 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(2*n-3+(4*n-6)*x)*P(n-1,x) + (n-2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 3 + (4*n - 6)*x)*P[n - 1, x] + (n - 2)*P[n - 2, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(2*n - 3 + (4*n - 6)*x)*v[i - 1] + (n - 2)*v[i - 2])/(-n + 1)); 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":"A063007","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: 1,  2;\n  2: 1,  6,    6;\n  3: 1, 12,   30,    20;\n  4: 1, 20,   90,   140,    70;\n  5: 1, 30,  210,   560,   630,    252;\n  6: 1, 42,  420,  1680,  3150,   2772,    924;\n  7: 1, 56,  756,  4200, 11550,  16632,  12012,   3432;\n  8: 1, 72, 1260,  9240, 34650,  72072,  84084,  51480,  12870;\n  9: 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;\n","recurrence_sentence":"The row polynomials satisfy -n*P(n,x) = -(2*n-1+(4*n-2)*x)*P(n-1,x) + (n-1)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (4*n - 2)*x)*P[n - 1, x] + (n - 1)*P[n - 2, x])/(-n)]];\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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1 + (4*n - 2)*x)*v[i - 1] + (n - 1)*v[i - 2])/(-n)); 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":"A104684","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:     2,      1;\n  2:     6,      6,      1;\n  3:    20,     30,     12,      1;\n  4:    70,    140,     90,     20,      1;\n  5:   252,    630,    560,    210,     30,     1;\n  6:   924,   2772,   3150,   1680,    420,    42,     1;\n  7:  3432,  12012,  16632,  11550,   4200,   756,    56,    1;\n  8: 12870,  51480,  84084,  72072,  34650,  9240,  1260,   72,  1;\n  9: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where -n*P(n,x) = -(2*n-1+(4*n-2)*x)*P(n-1,x) + (n-1)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (4*n - 2)*x)*P[n - 1, x] + (n - 1)*P[n - 2, x])/(-n)]];\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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1 + (4*n - 2)*x)*v[i - 1] + (n - 1)*v[i - 2])/(-n)); 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}}],"derivation":{"root":"n-4689fb1386d440ad","steps":[],"edges":[],"text":"A063007"},"headline":"A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","chain":null}