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A156691 Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(i+1) ) and m = 2, read by rows. 8
1, 1, 1, 1, -5, 1, 1, 40, 40, 1, 1, -440, 3520, -440, 1, 1, 6160, 542080, 542080, 6160, 1, 1, -104720, 129015040, -1419165440, 129015040, -104720, 1, 1, 2094400, 43865113600, 6755227494400, 6755227494400, 43865113600, 2094400, 1, 1, -48171200, 20177952256000, -52825879006208000, 739562306086912000, -52825879006208000, 20177952256000, -48171200, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, -3, 82, 2642, 1096482, -1161344798, 13598189404802, 633950903882665602, 301999235305843794118402, ...}.

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(i+1) ) and m = 2.

T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (2, 1, 1). - G. C. Greubel, Feb 25 2021

EXAMPLE

Triangle begins as:

1;

1, 1;

1, -5, 1;

1, 40, 40, 1;

1, -440, 3520, -440, 1;

1, 6160, 542080, 542080, 6160, 1;

1, -104720, 129015040, -1419165440, 129015040, -104720, 1;

1, 2094400, 43865113600, 6755227494400, 6755227494400, 43865113600, 2094400, 1;

MATHEMATICA

(* First program *)

t[n_, k_]:= If[k==0, n!, Product[1 -(i+1)*(k+1), {j, n}, {i, 0, j-1}] ];

T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)

(* Second program *)

f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];

T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));

Table[T[n, k, 2, 1, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

PROG

(Sage)

@CachedFunction

def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))

flatten([[T(n, k, 2, 1, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 25 2021

(Magma)

f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;

[T(n, k, 2, 1, 1): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A007318 (m=0), A156690 (m=1), this sequence (m=2), A156692 (m=3).

Cf. A156693, A156697, A156725.

Sequence in context: A015116 A322220 A174790 * A246051 A111820 A174912

Adjacent sequences: A156688 A156689 A156690 * A156692 A156693 A156694

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 13 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 25 2021

STATUS

approved

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Last modified January 27 20:03 EST 2023. Contains 359849 sequences. (Running on oeis4.)