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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + 2*n*x)*v[i - 1] - (1 + 2*x + x^2)*deriv(v[i - 1], x)); 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":"A059374","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:        3,         2;\n   3:       13,        18,          6;\n   4:       73,       156,        108,         24;\n   5:      501,      1460,       1560,        720,        120;\n   6:     4051,     15030,      21900,      15600,       5400,        720;\n   7:    37633,    170142,     315630,     306600,     163800,      45360,      5040;\n   8:   394353,   2107448,    4763976,    5891760,    4292400,    1834560,    423360,     40320;\n   9:  4596553,  28393416,   75868128,  114335424,  106051680,   61810560,  22014720,   4354560,   362880;\n  10: 58941091, 413689770, 1277703720, 2276043840, 2572547040, 1908930240, 927158400, 283046400, 48988800, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(2*n-2)*x)*P(n-1,x) - (1+2*x+x^2)*d/dx 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*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], 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] = (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (1 + 2*x + x^2)*deriv(v[i - 1], 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-d55979c30b97975a","steps":["x \u21a6 x+1"],"edges":["e-cee0a9f908e74de4"],"text":"A089231 \u2192 x \u21a6 x+1"},"headline":"A059374: Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","chain":"A089231 \u2192 x \u21a6 x+1"}