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\u21a6 x+1"},"headline":"A034867 \u2192 x \u21a6 x+1","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:    0;\n  1:   -1;\n  2:    2;\n  3:   -4,   -1;\n  4:    8,    4;\n  5:  -16,  -12,   -1;\n  6:   32,   32,    6;\n  7:  -64,  -80,  -24,  -1;\n  8:  128,  192,   80,   8;\n  9: -256, -448, -240, -40, -1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(2*n-2+(n-2)*x)*P(n-1,x) + (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = -1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := -1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (n - 2)*x)*P[n - 1, x] + (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 2 + (n - 2)*x)*v[i - 1] + (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-74c1e94669ff21ff","steps":["Veronese 2 (j=1)","basis A007318"],"edges":["e-a70814c31408551f","e-6264503eda6bd957"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 basis A007318"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 basis A007318","chain":null}