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A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1). 4
1, 2, 1, 8, 6, 1, 64, 56, 14, 1, 1024, 960, 280, 30, 1, 32768, 31744, 9920, 1240, 62, 1, 2097152, 2064384, 666624, 89280, 5208, 126, 1, 268435456, 266338304, 87392256, 12094464, 755904, 21336, 254, 1, 68719476736, 68451041280, 22638755840, 3183575040, 205605888, 6217920, 86360, 510, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [[1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006
LINKS
FORMULA
Sum_{k=0..n} T(n, k) = A028362(n).
T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - Philippe Deléham, Nov 05 2006
T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - Werner Schulte, Mar 25 2019
EXAMPLE
Triangle begins:
1;
2, 1;
8, 6, 1;
64, 56, 14, 1;
1024, 960, 280, 30, 1;
32768, 31744, 9920, 1240, 62, 1;
MATHEMATICA
T[n_, k_, q_]:= T[n, k, q]= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1, k, q] +T[n-1, k-1, q] ]];
Table[T[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)
PROG
(Sage)
def T(n, k, q):
if (k<0 or k>n): return 0
elif (k==n): return 1
else: return q^n*T(n-1, k, q) + T(n-1, k-1, q)
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021
(Magma)
function T(n, k, q)
if k lt 0 or k gt n then return 0;
elif k eq n then return 1;
else return q^n*T(n-1, k, q) + T(n-1, k-1, q);
end if; return T; end function;
[T(n, k, 2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021
CROSSREFS
Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).
Sequence in context: A039683 A318389 A108085 * A305860 A272983 A195698
KEYWORD
nonn,tabl
AUTHOR
Gerald McGarvey, Jun 05 2005
STATUS
approved

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