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A176263 Triangle T(n,k) = A015440(k) - A015440(n) + A015440(n-k), read by rows. 2
1, 1, 1, 1, -4, 1, 1, -4, -4, 1, 1, -29, -29, -29, 1, 1, -54, -79, -79, -54, 1, 1, -204, -254, -279, -254, -204, 1, 1, -479, -679, -729, -729, -679, -479, 1, 1, -1504, -1979, -2179, -2204, -2179, -1979, -1504, 1, 1, -3904, -5404, -5879, -6054, -6054, -5879, -5404, -3904, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Row sums are s(n) = {1, 2, -2, -6, -85, -264, -1193, -3772, -13526, -42480, -139159, ...}, obeying s(n) = 3*s(n-1) + 7*s(n-2) - 19*s(n-3) - 15*s(n-4) + 25*s(n-5) with g.f. (1-x-15*x^2+5*x^3)/((1-x)*(1-x-5*x^2)^2).
LINKS
EXAMPLE
Triangle begins as:
1;
1, 1;
1, -4, 1;
1, -4, -4, 1;
1, -29, -29, -29, 1;
1, -54, -79, -79, -54, 1;
1, -204, -254, -279, -254, -204, 1;
1, -479, -679, -729, -729, -679, -479, 1;
1, -1504, -1979, -2179, -2204, -2179, -1979, -1504, 1;
1, -3904, -5404, -5879, -6054, -6054, -5879, -5404, -3904, 1;
MAPLE
A176263 := proc(n, k)
A015440(k)-A015440(n)+A015440(n-k) ;
end proc; # R. J. Mathar, May 03 2013
MATHEMATICA
(* Set of sequences q=0..10 *)
f[n_, q_]:= f[n, q] = If[n<2, n, f[n-1, q] + q*f[n-2, q]];
T[n_, k_, q_]:= f[k+1, q] + f[n-k+1, q] - f[n+1, q];
Table[Flatten[Table[T[n, k, q], {n, 0, 10}, {k, 0, n}], {q, 0, 10}]
(* Second program *)
A015440[n_]:= Sum[5^j*Binomial[n-j, j], {j, 0, (n+1)/2}]; T[n_, k_]:= T[n, k]= A015440[k] +A015440[n-k] -A015440[n]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 24 2019 *)
PROG
(PARI) A015440(n) = sum(j=0, (n+1)\2, 5^j*binomial(n-j, j));
T(n, k) = A015440(k) - A015440(n) + A015440(n-k); \\ G. C. Greubel, Nov 24 2019
(Magma) A015440:= func< n | &+[5^j*Binomial(n-j, j): j in [0..Floor(n/2)]] >;
[A015440(k) - A015440(n) + A015440(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 24 2019
(Sage)
def A015440(n): return sum(5^j*binomial(n-j, j) for j in (0..floor(n/2)))
[[A015440(k) - A015440(n) + A015440(n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 24 2019
CROSSREFS
Sequence in context: A298575 A326039 A183374 * A110812 A151904 A356296
KEYWORD
sign,tabl,easy
AUTHOR
Roger L. Bagula, Apr 13 2010
STATUS
approved

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)