A302016 -id:A302016 - cp's OEIS frontend

A291678 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 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))).

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, -1, 0, 1, 4, 3, -2, 0, 0, 1, 5, 6, -2, -2, 1, 0, 1, 6, 10, 0, -6, 2, 1, 0, 1, 7, 15, 5, -11, 0, 5, -1, 0, 1, 8, 21, 14, -15, -8, 12, 0, -2, 0, 1, 9, 28, 28, -15, -24, 18, 9, -8, 0, 0, 1, 10, 36, 48, -7, -48, 15, 32, -15, -6, 2

Offset: 0

Seiichi Manyama, Aug 29 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   0,  1,  2,  3,   4, ...
   0,  0,  1,  3,   6, ...
   0, -1, -2, -2,   0, ...
   0,  0, -2, -6, -11, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000007, A003823, A285442, A285443, A285444.
Rows n=0..1 give A000012, A001477.
Main diagonal gives A291679.
Antidiagonal sums give A302016.
Cf. A286509.

G.f. of column k: Product_{j>=1} ((1 - x^(5*j-2))*(1 - x^(5*j-3)) / ((1 - x^(5*j-1))*(1 - x^(5*j-4))))^k.

A302015 Expansion of 1/(1 - x/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 0, 1, 0, -1, 1, 1, -2, -1, 2, 0, -2, 2, 3, -3, -3, 4, 0, -7, 3, 9, -5, -7, 10, 4, -17, -1, 21, -7, -21, 21, 19, -36, -13, 47, -5, -56, 36, 64, -69, -54, 104, 15, -147, 41, 177, -115, -168, 221, 116, -344, -15, 442, -159, -481, 422, 443, -736, -280, 1034, -90, -1276, 681

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Antidiagonal sums of A286509.

Cf. A007325, A167750, A302016.

nmax = 68; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[1/(1 - x QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5])), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4))/((1 - x^(5*k-2))*(1 - x^(5*k-3)))).

a(0) = 1; a(n) = Sum_{k=1..n} A007325(k-1)*a(n-k).

cp's OEIS Frontend

A291678 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 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))).

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