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begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  3,   2;\n  3: 1,  6,  11,    6;\n  4: 1, 10,  35,   50,    24;\n  5: 1, 15,  85,  225,   274,    120;\n  6: 1, 21, 175,  735,  1624,   1764,    720;\n  7: 1, 28, 322, 1960,  6769,  13132,  13068,    5040;\n  8: 1, 36, 546, 4536, 22449,  67284, 118124,  109584,   40320;\n  9: 1, 45, 870, 9450, 63273, 269325, 723680, 1172700, 1026576, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (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[(1 + n*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {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] = (1 + n*x)*v[i - 1]); 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":"A094638","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,   2;\n   4: 1,  6,  11,    6;\n   5: 1, 10,  35,   50,    24;\n   6: 1, 15,  85,  225,   274,    120;\n   7: 1, 21, 175,  735,  1624,   1764,    720;\n   8: 1, 28, 322, 1960,  6769,  13132,  13068,    5040;\n   9: 1, 36, 546, 4536, 22449,  67284, 118124,  109584,   40320;\n  10: 1, 45, 870, 9450, 63273, 269325, 723680, 1172700, 1026576, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 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":"A130534","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:      2,       3,       1;\n  3:      6,      11,       6,      1;\n  4:     24,      50,      35,     10,      1;\n  5:    120,     274,     225,     85,     15,     1;\n  6:    720,    1764,    1624,    735,    175,    21,    1;\n  7:   5040,   13068,   13132,   6769,   1960,   322,   28,   1;\n  8:  40320,  109584,  118124,  67284,  22449,  4536,  546,  36,  1;\n  9: 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (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[(1 + n*x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1]); 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-4d60c3dd8627c01d","steps":[],"edges":[],"text":"A094638"},"headline":"A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","chain":null}