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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A136124","target":"triangle","first_row":0,"oeis_offset":2,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   2:       1;\n   3:       2,       1;\n   4:       6,       5,       1;\n   5:      24,      26,       9,       1;\n   6:     120,     154,      71,      14,      1;\n   7:     720,    1044,     580,     155,     20,      1;\n   8:    5040,    8028,    5104,    1665,    295,     27,     1;\n   9:   40320,   69264,   48860,   18424,   4025,    511,    35,    1;\n  10:  362880,  663696,  509004,  214676,  54649,   8624,   826,   44,  1;\n  11: 3628800, 6999840, 5753736, 2655764, 761166, 140889, 16884, 1266, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+x)*P(n-1,x) for n >= 3, with P(2,x) = 1.","mathematica":"P[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten","pari":"P(N) = {my(v = vector(N - 1)); v[1] = 1; for(n = 3, N, my(i = n - 1); v[i] = (n - 1 + 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(11)","reading":{"first_row":0,"offset":2,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A143491","target":"triangle","first_row":0,"oeis_offset":2,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   2:       1;\n   3:       2,       1;\n   4:       6,       5,       1;\n   5:      24,      26,       9,       1;\n   6:     120,     154,      71,      14,      1;\n   7:     720,    1044,     580,     155,     20,      1;\n   8:    5040,    8028,    5104,    1665,    295,     27,     1;\n   9:   40320,   69264,   48860,   18424,   4025,    511,    35,    1;\n  10:  362880,  663696,  509004,  214676,  54649,   8624,   826,   44,  1;\n  11: 3628800, 6999840, 5753736, 2655764, 761166, 140889, 16884, 1266, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+x)*P(n-1,x) for n >= 3, with P(2,x) = 1.","mathematica":"P[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten","pari":"P(N) = {my(v = vector(N - 1)); v[1] = 1; for(n = 3, N, my(i = n - 1); v[i] = (n - 1 + 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(11)","reading":{"first_row":0,"offset":2,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A145324","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  5,    6;\n   4: 1,  9,   26,    24;\n   5: 1, 14,   71,   154,    120;\n   6: 1, 20,  155,   580,   1044,    720;\n   7: 1, 27,  295,  1665,   5104,   8028,    5040;\n   8: 1, 35,  511,  4025,  18424,  48860,   69264,   40320;\n   9: 1, 44,  826,  8624,  54649, 214676,  509004,  663696,  362880;\n  10: 1, 54, 1266, 16884, 140889, 761166, 2655764, 5753736, 6999840, 3628800;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+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[(n + x)*P[n - 1, x]];\nTable[Reverse[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] = (n + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-914700b1060feeb2","steps":[],"edges":[],"text":"A136124"},"headline":"A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","chain":null}