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133, 91, 18, 1;\n 3: 216, 1981, 3235, 1728, 371, 33, 1;\n 4: 1296, 27265, 89215, 94590, 42765, 9306, 1010, 52, 1;\n 5: 7776, 363517, 2209411, 4130100, 3286075, 1303731, 279747, 33540, 2220, 75, 1;\n 6: 46656, 4780153, 51886471, 160960338, 207060161, 132948816, 47269768, 9872304, 1246047, 95175, 4261, 102, 1;\n 7: 279936, 62468581, 1184570395, 5887222488, 11669138271, 11376740529, 6154231937, 1988687896, 401621698, 51951060, 4332748, 229908, 7441, 133, 1;\n 8: 1679616, 814051105, 26623045855, 207371085030, 614498823205, 874905468822, 685103818350, 321209873244, 95308038117, 18551086410, 2419634366, 213307500, 12651982, 494228, 12116, 168, 1;\n 9: 10077696, 10594421677, 593035148371, 7136913361500, 30989393018675, 62794087031247, 68925989624479, 45049235722140, 18676859654280, 5128997933745, 960744503123, 125049561792, 11414291870, 730558626, 32426682, 972216, 18690, 207, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+7*x+x^2)*P(n-1,x) + (7*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 7*x + x^2)*P[n - 1, x] + (7*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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] = (6 + 7*x + x^2)*v[i - 1] + (7*x + 2*x^2)*deriv(v[i - 1], x) + x^2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d3c3090977105807","steps":["basis A008277"],"edges":["e-efe78907c9beaa2c"],"text":"A272867 \u2192 basis A008277"},"headline":"A272867 \u2192 basis A008277","chain":null}