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%I #42 Apr 15 2021 23:48:08
%S 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,
%T 1050,105,1,1,168,2940,14700,14700,2940,168,1,1,252,7056,61740,123480,
%U 61740,7056,252,1,1,360,15120,211680,740880,740880,211680,15120,360,1
%N 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.
%C 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.
%D T. S. Blyth and E. F. Robertson, Essential Student Algebra: Groups, Volume 5, J. W. Arrowsmith, Bristol, 1986, page 14.
%D Leonard Eugene Dickson, On Invariants and the Theory of Numbers, Dover, New York, 1966, page 34.
%H G. C. Greubel, <a href="/A174150/b174150.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Leonard Eugene Dickson, <a href="https://archive.org/details/madisoncolloquiu00amer/page/34">On Invariants and the Theory of Numbers</a>, American Mathematical Society, Colloquium Lectures, 1913, p. 34.
%F 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.
%F 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
%e Triangle T(n,m) (with rows n >= 0 and columns m >= 0) begins as follows:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 12, 12, 1;
%e 1, 30, 60, 30, 1;
%e 1, 60, 300, 300, 60, 1;
%e 1, 105, 1050, 2625, 1050, 105, 1;
%e 1, 168, 2940, 14700, 14700, 2940, 168, 1;
%e 1, 252, 7056, 61740, 123480, 61740, 7056, 252, 1;
%e 1, 360, 15120, 211680, 740880, 740880, 211680, 15120, 360, 1;
%e 1, 495, 29700, 623700, 3492720, 6112260, 3492720, 623700, 29700, 495, 1;
%e ...
%e Row sums are: 1, 2, 8, 26, 122, 722, 4937, 35618, 261578, 1936082, 14405492, ...
%p g := n -> `if`(n < 2, 1, GAMMA(n)*GAMMA(n+1)*GAMMA(n+2)/2^(n-1)):
%p T := (n, k) -> round(g(n)/(g(k)*g(n-k))):
%p seq(seq(T(n, k), k=0..n), n=0..12); # _Peter Luschny_, Sep 02 2019
%t (* First program *)
%t c[n_]:= If[n<2, 1, 2^(2-n)*Product[i*(i^2 -1), {i,2,n}]];
%t T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%t (* Second program *)
%t 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;
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 15 2021 *)
%o (Magma)
%o function t(n,k)
%o if k eq 0 or k eq n then return 1;
%o elif k eq 1 and n eq 2 then return 6;
%o elif k eq 1 or k eq n-1 then return 3*Binomial(n+1,3);
%o else return (k/(2*(k+1)))*(&*[Binomial(n+j-1,k): j in [0..2]]);
%o end if; return t;
%o end function;
%o [Round(t(n,k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 15 2021
%o (Sage)
%o @CachedFunction
%o def t(n, k):
%o if (k==0 or k==n): return 1
%o elif (k==1 and n==2): return 6
%o elif (k==1 or k==n-1): return 3*binomial(n+1,3)
%o else: return (k/(2*(k+1)))*product(binomial(n+j-1,k) for j in (0..2))
%o flatten([[t(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 15 2021
%Y Cf. A174151.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 10 2010
%E Edited and renamed by _Peter Luschny_, Sep 02 2019
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