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n); v[i] = ((2*n^2 + n + (13*n^2 - 23*n + 4)*x)*v[i - 1] + ((4*n - 1)*x - ((27/2)*n - 3)*x^2)*deriv(v[i - 1], x) + ((5/2)*n^2 - 6*n + 2)*x^2*v[i - 2])/(2*n^2 + n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A391045","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  4,    1;\n  3: 1,  9,   10,     1;\n  4: 1, 16,   42,    20,     1;\n  5: 1, 25,  120,   140,    35,     1;\n  6: 1, 36,  275,   600,   378,    56,     1;\n  7: 1, 49,  546,  1925,  2310,   882,    84,    1;\n  8: 1, 64,  980,  5096, 10010,  7392,  1848,  120,   1;\n  9: 1, 81, 1632, 11760, 34398, 42042, 20592, 3564, 165, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (2*n^2+5*n+3+(13*n^2+3*n-6)*x)*P(n-1,x) + ((4*n+3)*x-((27/2)*n+21/2)*x^2)*d/dx P(n-1,x) + ((5/2)*n^2-n-3/2)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 5*n + 3 + (13*n^2 + 3*n - 6)*x)*P[n - 1, x] + ((4*n + 3)*x - ((27/2)*n + 21/2)*x^2)*D[P[n - 1, x], x] + ((5/2)*n^2 - n - 3/2)*x^2*P[n - 2, x])/(2*n^2 + 5*n + 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 5*n + 3 + (13*n^2 + 3*n - 6)*x)*v[i - 1] + ((4*n + 3)*x - ((27/2)*n + 21/2)*x^2)*deriv(v[i - 1], x) + ((5/2)*n^2 - n - 3/2)*x^2*v[i - 2])/(2*n^2 + 5*n + 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-79ebafb266da3b41","steps":["\u2299 C(n,k+1) (shift -1)"],"edges":["e-1491609a6fc5f8c9"],"text":"A172431 \u2192 \u2299 C(n,k+1) (shift -1)"},"headline":"A391045: Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","chain":"A172431 \u2192 \u2299 C(n,k+1) (shift -1)"}