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1;\n  1:       3,           7,              2;\n  2:      13,         135,            187,               66,                 6;\n  3:      63,        2377,           9026,             9071,              3262,               450,                 20;\n  4:     321,       41935,         375905,           814850,            646471,            224370,              36350,               2660,                70;\n  5:    1683,      751117,       14966132,         62934125,          93083985,          61149985,           20093754,            3486630,            319970,             14490,              252;\n  6:    8989,    13651491,      588355495,       4554882584,       11613156884,       12886911555,         7231826155,         2234761494,         397640418,          41237490,          2432346,            74844,             924;\n  7:   48639,   251064961,    23106104618,     320013507051,     1346615653698,     2376908518250,      2097922138564,      1028147108371,      298309557830,       53174249730,       5912352180,        407158290,        16725324,         372372,         3432;\n  8:  265729,  4660302591,   910288233889,   22184328433290,   150066446050371,   406374021639978,    537942408789954,    392801303472516,   170917111295971,    46594624435266,    8209143592854,     949361546364,     72109283862,     3535023492,    106905084,     1801800,     12870;\n  9: 1462563, 87138646237, 36018966955292, 1529648141313605, 16357340665435285, 66386287592022213, 127863864211009526, 133852422505806930, 83168104891674705, 32546665634639201, 8351370659086666, 1442564948310750, 170259121838410, 13787972368170, 760244725812, 27872807820, 646352850, 8532810, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n-1+(3*n-3)*x+(n-1)*x^2)*P(n-2,x) - ((3*n-3)*x+(2*n-2)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (6*n-3+(14*n-7)*x+(4*n-2)*x^2)*P(n-1,x) + ((14*n-7)*x+(8*n-4)*x^2)*d/dx P(n-1,x) + (4*n-2)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 7*x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 7*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (3*n - 3)*x + (n - 1)*x^2)*P[n - 2, x] - ((3*n - 3)*x + (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (6*n - 3 + (14*n - 7)*x + (4*n - 2)*x^2)*P[n - 1, x] + ((14*n - 7)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x] + (4*n - 2)*x^2*D[P[n - 1, x], {x, 2}])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 7*x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (3*n - 3)*x + (n - 1)*x^2)*v[i - 2] - ((3*n - 3)*x + (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (6*n - 3 + (14*n - 7)*x + (4*n - 2)*x^2)*v[i - 1] + ((14*n - 7)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x) + (4*n - 2)*x^2*deriv(deriv(v[i - 1], x), x))/(n)); 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-e4d248920a317259","steps":["basis A008277"],"edges":["e-9db85d58368a545b"],"text":"A109983 \u2192 basis A008277"},"headline":"A109983 \u2192 basis A008277","chain":null}