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A156693 Triangle T(n, k) = Product_{j=1..k} Product_{i=0..j-1} ( 1 - (n-k+1)*(i+1) ) with T(n, 0) = 1 and T(n, n) = n!, read by rows. 8
1, 1, 1, 1, -1, 2, 1, -2, -3, 6, 1, -3, -20, 45, 24, 1, -4, -63, 1600, 4725, 120, 1, -5, -144, 14553, 1408000, -4465125, 720, 1, -6, -275, 72576, 50426145, -17346560000, -46414974375, 5040, 1, -7, -468, 257125, 694987776, -3319805256075, -3633063526400000, 6272287562165625, 40320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row sums are: {1, 2, 2, 2, 47, 6379, -3042000, -63711030894, 2635904925794297,

27927233169645980904169, 1028241148994588972886080924800, ...}.

LINKS

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

FORMULA

Let the square array t(n, k) be given by t(n, k) = Product_{j=1..n} Product_{i=0..j-1} ( 1 - (k+1)*(i +1) ) with t(n, 0) = n!. The number triangle, T(n, k), is the downward antidiagonals, i.e. T(n, k) = t(k, n-k).

T(n, k) = (-(n-k+1))^binomial(k+1, 2)*Product_{j=1..k} Pochhammer( (n-k)/(n-k+1), j) with T(n, 0) = 1 and T(n, n) = n!. - G. C. Greubel, Feb 25 2021

EXAMPLE

Triangle begins as:

1;

1, 1;

1, -1, 2;

1, -2, -3, 6;

1, -3, -20, 45, 24;

1, -4, -63, 1600, 4725, 120;

1, -5, -144, 14553, 1408000, -4465125, 720;

1, -6, -275, 72576, 50426145, -17346560000, -46414974375, 5040;

MATHEMATICA

(* First program *)

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

Table[t[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)

(* Second program *)

T[n_, k_, p_, q_]:= If[k==0, 1, If[k==n, n!, (-p*(n-k+1))^Binomial[k+1, 2]*Product[ Pochhammer[(q*(n-k+1) -1)/(p*(n-k+1)), j], {j, k}]]];

Table[T[n, k, 1, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

PROG

(Sage)

@CachedFunction

def T(n, k, p, q):

if (k==0): return 1

elif (k==n): return factorial(n)

else: return (-p*(n-k+1))^binomial(k+1, 2)*product( rising_factorial( (q*(n-k+1)-1)/(p*(n-k+1)), j) for j in (1..k))

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

(Magma)

function T(n, k, p, q)

if k eq 0 then return 1;

elif k eq n then return Factorial(n);

else return (&*[1 - (n-k+1)*(p*m+q): m in [0..j-1], j in [1..k]]);

end if; return T;

end function;

[T(n, k, 1, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A156690, A156691, A156692.

Cf. A156699, A156730.

Sequence in context: A086582 A033639 A156565 * A156564 A198123 A106576

Adjacent sequences: A156690 A156691 A156692 * A156694 A156695 A156696

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 February 2 06:48 EST 2023. Contains 360000 sequences. (Running on oeis4.)