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A156576 Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (i+1)*(k+1)^i ) with T(n, 0) = n!, read by antidiagonals. 1
1, 1, 1, 1, 1, 2, 1, 1, 5, 6, 1, 1, 7, 85, 24, 1, 1, 9, 238, 4165, 120, 1, 1, 11, 513, 33796, 537285, 720, 1, 1, 13, 946, 160569, 18486412, 172468485, 5040, 1, 1, 15, 1573, 554356, 255786417, 37065256060, 132628264965, 40320, 1, 1, 17, 2430, 1549405, 2057215116, 1979019508329, 263459840074480, 237802479082245, 362880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

FORMULA

T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (i+1)*(k+1)^i ) with T(n, 0) = n! (square array).

T(n, k) = (1/k^(2*n))*Product_{j=1..n} (1 -(j+1)*(k+1)^j +j*(k+1)^(j+1)) with T(n, 0) = n! (square array). - G. C. Greubel, Jun 28 2021

EXAMPLE

Square array begins as:

    1,      1,        1,         1,          1,           1 ...;

    1,      1,        1,         1,          1,           1 ...;

    2,      5,        7,         9,         11,          13 ...;

    6,     85,      238,       513,        946,        1573 ...;

   24,   4165,    33796,    160569,     554356,     1549405 ...;

  120, 537285, 18486412, 255786417, 2057215116, 11566308325 ...;

Antidiagonal triangle begins as:

  1;

  1, 1;

  1, 1,  2;

  1, 1,  5,    6;

  1, 1,  7,   85,     24;

  1, 1,  9,  238,   4165,       120;

  1, 1, 11,  513,  33796,    537285,         720;

  1, 1, 13,  946, 160569,  18486412,   172468485,         5040;

  1, 1, 15, 1573, 554356, 255786417, 37065256060, 132628264965, 40320;

MATHEMATICA

(* First program *)

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

Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 28 2021 *)

(* Second program *)

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

Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)

PROG

(MAGMA)

A156576:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (1/k^(2*n))*(&*[1 -(j+1)*(k+1)^j +j*(k+1)^(j+1): j in [1..n]]) >;

[A156576(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

(Sage)

def A156576(n, k): return factorial(n) if (k==0) else (1/k^(2*n))*product( 1 -(j+1)*(k+1)^j +j*(k+1)^(j+1) for j in [1..n])

flatten([[A156576(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

CROSSREFS

Sequence in context: A086856 A052916 A326048 * A293219 A266572 A266681

Adjacent sequences:  A156573 A156574 A156575 * A156577 A156578 A156579

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 10 2009

EXTENSIONS

Edited by G. C. Greubel, Jun 28 2021

STATUS

approved

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Last modified September 20 05:17 EDT 2021. Contains 347577 sequences. (Running on oeis4.)