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v[1] = x; for(n = 2, N, my(i = n); v[i] = (n - 1 + n*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":1,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A088996","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,     1;\n  2: 0,     1,      2;\n  3: 0,     2,      7,       6;\n  4: 0,     6,     29,      46,      24;\n  5: 0,    24,    146,     329,     326,     120;\n  6: 0,   120,    874,    2521,    3604,    2556,     720;\n  7: 0,   720,   6084,   21244,   39271,   40564,   22212,    5040;\n  8: 0,  5040,  48348,  197380,  444849,  598116,  479996,  212976,   40320;\n  9: 0, 40320, 432144, 2014172, 5335212, 8788569, 9223012, 6023772, 2239344, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+n*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + n*x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (n - 1 + n*x)*v[i - 1]); 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-849de7332df28657","steps":[],"edges":[],"text":"A088996"},"headline":"A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","chain":null}