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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 (list; table; graph; refs; listen; history; text; internal format)
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

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Adjacent sequences:  A174147 A174148 A174149 * A174151 A174152 A174153

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Mar 10 2010

EXTENSIONS

Edited and renamed by Peter Luschny, Sep 02 2019

STATUS

approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)