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begins:\n  0: 1;\n  1: 1,      3,         1;\n  2: 1,     15,        25,          10,            1;\n  3: 1,     63,       301,         350,          140,           21,             1;\n  4: 1,    255,      3025,        7770,         6951,         2646,           462,            36,             1;\n  5: 1,   1023,     28501,      145750,       246730,       179487,         63987,         11880,          1155,           55,            1;\n  6: 1,   4095,    261625,     2532530,      7508501,      9321312,       5715424,       1899612,        359502,        39325,         2431,          78,          1;\n  7: 1,  16383,   2375101,    42355950,    210766920,    420693273,     408741333,     216627840,      67128490,     12662650,      1479478,      106470,       4550,       105,        1;\n  8: 1,  65535,  21457825,   694337290,   5652751651,  17505749898,   25708104786,   20415995028,    9528822303,   2758334150,    512060978,    62022324,    4910178,    249900,     7820,    136,     1;\n  9: 1, 262143, 193448101, 11259666950, 147589284710, 693081601779, 1492924634839, 1709751003480, 1144614626805, 477297033785, 129413217791, 23466951300, 2892439160, 243577530, 13916778, 527136, 12597, 171, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-1,x) + (3*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[(1 + 3*x + x^2)*P[n - 1, x] + (3*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] = (1 + 3*x + x^2)*v[i - 1] + (3*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-cdb0db4ad473ee28","steps":["basis A008277"],"edges":["e-7ccdf7ec12150e18"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008277","chain":null}