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A174151 Triangle T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = 12*Product_{j=3..n} ( 12 * binomial(j+2, 4) ) with c(0) = c(1) = 1, read by rows. 2
1, 1, 1, 1, 12, 1, 1, 60, 60, 1, 1, 180, 900, 180, 1, 1, 420, 6300, 6300, 420, 1, 1, 840, 29400, 88200, 29400, 840, 1, 1, 1512, 105840, 740880, 740880, 105840, 1512, 1, 1, 2520, 317520, 4445280, 10372320, 4445280, 317520, 2520, 1, 1, 3960, 831600, 20956320, 97796160, 97796160, 20956320, 831600, 3960, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = 12*Product_{j=3..n} ( 12 * binomial(j+2, 4) ) with c(0) = c(1) = 1.
T(n, k) = (6*k/((k+1)^2*(k+2))*Product_{j=0..3} binomial(n+j-1, k) with T(n, 0) = T(n, n) = 1. - G. C. Greubel, Apr 16 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 12, 1;
1, 60, 60, 1;
1, 180, 900, 180, 1;
1, 420, 6300, 6300, 420, 1;
1, 840, 29400, 88200, 29400, 840, 1;
1, 1512, 105840, 740880, 740880, 105840, 1512, 1;
1, 2520, 317520, 4445280, 10372320, 4445280, 317520, 2520, 1;
1, 3960, 831600, 20956320, 97796160, 97796160, 20956320, 831600, 3960, 1;
MATHEMATICA
(* First program *)
c[n_]:= If[n<2, 1, 12*Product[i*(i^2-1)*(i+2)/2], {i, 3, n}]];
T[n_, m_] = c[n]/(c[m]*c[n-m]);
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, (6*k/((k+1)^2*(k+2)))*Product[Binomial[n+j-1, k], {j, 0, 3}] ];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 16 2021 *)
PROG
(Magma)
function T(n, k)
if k eq 0 or k eq n then return 1;
else return (6*k/((k+1)^2*(k+2)))*(&*[Binomial(n+j-1, k): j in [0..3]]);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 16 2021
(Sage)
@CachedFunction
def T(n, k):
if (k==0 or k==n): return 1
else: return (6*k/((k+1)^2*(k+2)))*product(binomial(n+j-1, k) for j in (0..3))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021
CROSSREFS
Cf. A174150.
Sequence in context: A168518 A174667 A174672 * A342890 A155491 A142460
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Mar 10 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 16 2021
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)