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begins:\n  0: 1;\n  1: 0,       2;\n  2: 0,       6,        12;\n  3: 0,      24,       120,        120;\n  4: 0,     120,      1080,       2520,        1680;\n  5: 0,     720,     10080,      40320,       60480,       30240;\n  6: 0,    5040,    100800,     604800,     1512000,     1663200,       665280;\n  7: 0,   40320,   1088640,    9072000,    33264000,    59875200,     51891840,     17297280;\n  8: 0,  362880,  12700800,  139708800,   698544000,  1816214400,   2542700160,   1816214400,   518918400;\n  9: 0, 3628800, 159667200, 2235340800, 14529715200, 50854003200, 101708006400, 116237721600, 70572902400, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(4*n-2)*x)*P(n-1,x) - (n^2-2*n)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n^2 - 2*n)*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n^2 - 2*n)*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":"A357367","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: 0,       2;\n  2: 0,       6,        12;\n  3: 0,      24,       120,        120;\n  4: 0,     120,      1080,       2520,        1680;\n  5: 0,     720,     10080,      40320,       60480,       30240;\n  6: 0,    5040,    100800,     604800,     1512000,     1663200,       665280;\n  7: 0,   40320,   1088640,    9072000,    33264000,    59875200,     51891840,     17297280;\n  8: 0,  362880,  12700800,  139708800,   698544000,  1816214400,   2542700160,   1816214400,   518918400;\n  9: 0, 3628800, 159667200, 2235340800, 14529715200, 50854003200, 101708006400, 116237721600, 70572902400, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(4*n-2)*x)*P(n-1,x) - (n^2-2*n)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n^2 - 2*n)*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n^2 - 2*n)*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-4cba07695b3e3f90","steps":[],"edges":[],"text":"A357367"},"headline":"A357367: Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.","chain":null}