A291652 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 5, 0, 1, 6, 15, 24, 27, 20, 9, 0, 1, 7, 21, 40, 55, 54, 38, 15, 0, 1, 8, 28, 62, 100, 120, 109, 70, 26, 0, 1, 9, 36, 91, 168, 236, 258, 216, 129, 45, 0, 1, 10, 45, 128, 266, 426, 540, 544, 423, 238, 78, 0, 1, 11, 55, 174, 402, 721, 1035, 1205, 1127, 824, 437, 135, 0
Offset: 0
Examples
G.f. of column k: A_k(x) = 1 + k*x + k*(k + 1)*x^2/2 + k*(k^2 + 3*k + 8)*x^3/6 + k*(k^3 + 6*k^2 + 35*k + 30)*x^4/24 + ... Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, ... 0, 1, 3, 6, 10, 15, ... 0, 2, 6, 13, 24, 40, ... 0, 3, 11, 27, 55, 100, ... 0, 5, 20, 54, 120, 236, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
- Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
Crossrefs
Programs
-
Mathematica
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-x^i, 1, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[((Sum[(-1)^i x^(i (i + 1))/Product[(1 - x^m), {m, 1, i}], {i, 0, n}])/(Sum[(-1)^i x^(i^2)/Product[(1 - x^m), {m, 1, i}], {i, 0, n}]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Formula
G.f. of column k: (1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))))^k, a continued fraction.