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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A154986","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: 1,   1;\n  2: 1,   4,     1;\n  3: 1,  11,    11,      1;\n  4: 1,  24,    70,     24,       1;\n  5: 1,  45,   314,    314,      45,       1;\n  6: 1,  76,  1079,   2728,    1079,      76,      1;\n  7: 1, 119,  3045,  16995,   16995,    3045,    119,     1;\n  8: 1, 176,  7420,  80464,  186758,   80464,   7420,   176,   1;\n  9: 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (n^2-n)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (n^2 - n)*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n^2 - n)*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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A154986","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,   1;\n  2: 1,   4,     1;\n  3: 1,  11,    11,      1;\n  4: 1,  24,    70,     24,       1;\n  5: 1,  45,   314,    314,      45,       1;\n  6: 1,  76,  1079,   2728,    1079,      76,      1;\n  7: 1, 119,  3045,  16995,   16995,    3045,    119,     1;\n  8: 1, 176,  7420,  80464,  186758,   80464,   7420,   176,   1;\n  9: 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (n^2-n)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (n^2 - n)*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n^2 - n)*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":0,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-08ac2c6d9914754c","steps":[],"edges":[],"text":"A154986"},"headline":"A154986: Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.","chain":null}