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(\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    6,    24;\n  2:   21,   168,    189;\n  3:   56,   672,   1512,     896;\n  4:  126,  2016,   6804,    8064,    3150;\n  5:  252,  5040,  22680,   40320,   31500,    9072;\n  6:  462, 11088,  62370,  147840,  173250,   99792,   22638;\n  7:  792, 22176, 149688,  443520,  693000,  598752,  271656,   50688;\n  8: 1287, 41184, 324324, 1153152, 2252250, 2594592, 1765764,  658944,  104247;\n  9: 2002, 72072, 648648, 2690688, 6306300, 9081072, 8240232, 4612608, 1459458, 200200;\n","recurrence_sentence":"The row polynomials satisfy (n+(3*n^2-n)*x+n^3*x^2)*P(n,x) = (n+5+(3*n^2+18*n+15)*x+(n^3+10*n^2+28*n+15)*x^2+(n^3+7*n^2+11*n+5)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 5 + (3*n^2 + 18*n + 15)*x + (n^3 + 10*n^2 + 28*n + 15)*x^2 + (n^3 + 7*n^2 + 11*n + 5)*x^3)*P[n - 1, x])/(n + (3*n^2 - n)*x + n^3*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] = ((n + 5 + (3*n^2 + 18*n + 15)*x + (n^3 + 10*n^2 + 28*n + 15)*x^2 + (n^3 + 7*n^2 + 11*n + 5)*x^3)*v[i - 1])/(n + (3*n^2 - n)*x + n^3*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}}],"derivation":{"root":"n-97228a88c81561dc","steps":["(\u03b8+1)\u00b2"],"edges":["e-ac2d87fdb4f86514"],"text":"A178822 \u2192 (\u03b8+1)\u00b2"},"headline":"A178822 \u2192 (\u03b8+1)\u00b2","chain":null}