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A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0. 6
1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
FORMULA
G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + 2 * x * A_k(x)).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
-2, -1, 0, 1, 2, 3, 4, ...
2, -1, -1, 2, 8, 17, 29, ...
4, 5, 0, 5, 36, 109, 240, ...
-24, -3, 2, 13, 177, 766, 2177, ...
48, -21, 0, 36, 922, 5699, 20910, ...
MATHEMATICA
T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
PROG
(PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, (-2)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
(PARI) {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+2*x*A)); polcoef(A, n)}
CROSSREFS
Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.
Main diagonal gives A336714.
Sequence in context: A340032 A327785 A105242 * A221362 A114116 A054532
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Aug 01 2020
STATUS
approved

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Last modified April 18 09:47 EDT 2024. Contains 371779 sequences. (Running on oeis4.)