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v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A089231","target":"reversed_triangle","first_row":1,"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,  6,    6;\n   4: 1, 12,   36,    24;\n   5: 1, 20,  120,   240,    120;\n   6: 1, 30,  300,  1200,   1800,     720;\n   7: 1, 42,  630,  4200,  12600,   15120,     5040;\n   8: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;\n   9: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;\n  10: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\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] = (2 + x)*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); 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":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A105278","target":"triangle","first_row":1,"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:       2,        1;\n   3:       6,        6,        1;\n   4:      24,       36,       12,        1;\n   5:     120,      240,      120,       20,       1;\n   6:     720,     1800,     1200,      300,      30,      1;\n   7:    5040,    15120,    12600,     4200,     630,     42,     1;\n   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;\n   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;\n  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\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] = (2 + x)*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); 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":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-4a0fd05ab0621175","steps":[],"edges":[],"text":"A105278"},"headline":"A105278: Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","chain":null}