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A174043
Triangle T(n, k, q) = Sum_{j=0..10} q^j * floor( binomial(n+1,k)*binomial(n-1,k-1)/(2^j*(n+1)) ) for q = 1, read by rows.
3
1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 18, 38, 18, 1, 1, 26, 97, 97, 26, 1, 1, 39, 206, 344, 206, 39, 1, 1, 53, 389, 974, 974, 389, 53, 1, 1, 70, 669, 2348, 3522, 2348, 669, 70, 1, 1, 86, 1076, 5033, 10575, 10575, 5033, 1076, 86, 1, 1, 105, 1644, 9890, 27704, 38784, 27704, 9890, 1644, 105, 1
OFFSET
1,5
COMMENTS
Row sums are: {1, 2, 6, 22, 76, 248, 836, 2834, 9698, 33542, ...}.
FORMULA
T(n, k, q) = Sum_{j=0..10} q^j * floor( binomial(n+1,k)*binomial(n-1,k-1)/(2^j*(n+1)) ) for q = 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 4, 1;
1, 10, 10, 1;
1, 18, 38, 18, 1;
1, 26, 97, 97, 26, 1;
1, 39, 206, 344, 206, 39, 1;
1, 53, 389, 974, 974, 389, 53, 1;
1, 70, 669, 2348, 3522, 2348, 669, 70, 1;
1, 86, 1076, 5033, 10575, 10575, 5033, 1076, 86, 1;
MATHEMATICA
T[n_, k_, q_]:= Sum[q^j*Floor[Binomial[n-1, k-1]*Binomial[n+1, k]/(2^j*(n+1))], {j, 0, 10}];
Table[T[n, k, 1], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 16 2021 *)
PROG
(Magma)
T:= func< n, k, q | (&+[ q^j*Floor( Binomial(n+1, k)*Binomial(n-1, k-1)/(2^j*(n+1)) ): j in [0..10]]) >;
[T(n, k, 1): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2021
(Sage)
def T(n, k, q): return sum( q^j*( (binomial(n+1, k)*binomial(n-1, k-1)//(2^j*(n+1))) ) for j in (0..10))
flatten([[T(n, k, 1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 16 2021
CROSSREFS
Cf. this sequence (q=1), A174044 (q=2), A174045 (q=3).
Sequence in context: A109955 A214398 A220860 * A319029 A175124 A089447
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Mar 06 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 16 2021
STATUS
approved