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A174150
Triangle T(n, k) = round(c(n)/(c(k)*c(n-k))) where c(n) = ((n-1)! * n! * (n+1)!)/ 2^(n-1) if n >= 2, otherwise 1, read by rows.
2
1, 1, 1, 1, 6, 1, 1, 12, 12, 1, 1, 30, 60, 30, 1, 1, 60, 300, 300, 60, 1, 1, 105, 1050, 2625, 1050, 105, 1, 1, 168, 2940, 14700, 14700, 2940, 168, 1, 1, 252, 7056, 61740, 123480, 61740, 7056, 252, 1, 1, 360, 15120, 211680, 740880, 740880, 211680, 15120, 360, 1
OFFSET
0,5
COMMENTS
Based on the SL(2,p) number of prime modular form group transforms: f(p) = p*(p^2-1) if p = 2, otherwise f(p) = p*(p^2-1)/2.
REFERENCES
T. S. Blyth and E. F. Robertson, Essential Student Algebra: Groups, Volume 5, J. W. Arrowsmith, Bristol, 1986, page 14.
Leonard Eugene Dickson, On Invariants and the Theory of Numbers, Dover, New York, 1966, page 34.
LINKS
Leonard Eugene Dickson, On Invariants and the Theory of Numbers, American Mathematical Society, Colloquium Lectures, 1913, p. 34.
FORMULA
T(n, k) = round(c(n)/(c(k)*c(n-k))) for n >= 0 and k >= 0, where c(n) = 2^(2-n)* Product_{j=2..n} j*(j^2 - 1) for n >= 2 and otherwise 1.
T(n, k) = (k/(2*(k+1)))*Product_{j=0..2} binomial(n+j-1,k) with T(n,0) = T(n,n) = 1, T(n,1) = T(n,n-1) = 3*binomial(n+1,3) + 3*[n=2]. - G. C. Greubel, Apr 15 2021
EXAMPLE
Triangle T(n,m) (with rows n >= 0 and columns m >= 0) begins as follows:
1;
1, 1;
1, 6, 1;
1, 12, 12, 1;
1, 30, 60, 30, 1;
1, 60, 300, 300, 60, 1;
1, 105, 1050, 2625, 1050, 105, 1;
1, 168, 2940, 14700, 14700, 2940, 168, 1;
1, 252, 7056, 61740, 123480, 61740, 7056, 252, 1;
1, 360, 15120, 211680, 740880, 740880, 211680, 15120, 360, 1;
1, 495, 29700, 623700, 3492720, 6112260, 3492720, 623700, 29700, 495, 1;
...
Row sums are: 1, 2, 8, 26, 122, 722, 4937, 35618, 261578, 1936082, 14405492, ...
MAPLE
g := n -> `if`(n < 2, 1, GAMMA(n)*GAMMA(n+1)*GAMMA(n+2)/2^(n-1)):
T := (n, k) -> round(g(n)/(g(k)*g(n-k))):
seq(seq(T(n, k), k=0..n), n=0..12); # Peter Luschny, Sep 02 2019
MATHEMATICA
(* First program *)
c[n_]:= If[n<2, 1, 2^(2-n)*Product[i*(i^2 -1), {i, 2, n}]];
T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
(* Second program *)
T[n_, k_]:= If[k==0 || k==n, 1, If[k==1 || k==n-1, 3*Binomial[n+1, 3] + 3*Boole[n==2], (k/(2*(k+1)))* Product[Binomial[n+j-1, k], {j, 0, 2}] ]]//Round;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 15 2021 *)
PROG
(Magma)
function t(n, k)
if k eq 0 or k eq n then return 1;
elif k eq 1 and n eq 2 then return 6;
elif k eq 1 or k eq n-1 then return 3*Binomial(n+1, 3);
else return (k/(2*(k+1)))*(&*[Binomial(n+j-1, k): j in [0..2]]);
end if; return t;
end function;
[Round(t(n, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 15 2021
(Sage)
@CachedFunction
def t(n, k):
if (k==0 or k==n): return 1
elif (k==1 and n==2): return 6
elif (k==1 or k==n-1): return 3*binomial(n+1, 3)
else: return (k/(2*(k+1)))*product(binomial(n+j-1, k) for j in (0..2))
flatten([[t(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 15 2021
CROSSREFS
Cf. A174151.
Sequence in context: A174124 A174345 A174449 * A202673 A202875 A203956
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Mar 10 2010
EXTENSIONS
Edited and renamed by Peter Luschny, Sep 02 2019
STATUS
approved