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v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + 4*x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2)); 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":"A086915","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:       2;\n   2:       4,        4;\n   3:      12,       24,         8;\n   4:      48,      144,        96,        16;\n   5:     240,      960,       960,       320,        32;\n   6:    1440,     7200,      9600,      4800,       960,       64;\n   7:   10080,    60480,    100800,     67200,     20160,     2688,     128;\n   8:   80640,   564480,   1128960,    940800,    376320,    75264,    7168,    256;\n   9:  725760,  5806080,  13547520,  13547520,   6773760,  1806336,  258048,  18432,   512;\n  10: 7257600, 65318400, 174182400, 203212800, 121927680, 40642560, 7741440, 829440, 46080, 1024;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (4+4*x)*P(n-1,x) + (2+4*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4 + 4*x)*P[n - 1, x] + (2 + 4*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((4 + 4*x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2)); 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-cbd7c2e66114e926","steps":[],"edges":[],"text":"A086915"},"headline":"A086915: Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","chain":null}