A076111 -id:A076111 - cp's OEIS frontend

A256268 Table of k-fold factorials, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 4, 1, 1, 1, 120, 105, 28, 5, 1, 1, 1, 720, 945, 280, 45, 6, 1, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1

Offset: 0

Philippe Deléham, Jun 01 2015

A variant of A142589.

			1  1   1    1     1       1         1... A000012
1  1   2    6    24     120       720... A000142
1  1   3   15   105     945     10395... A001147
1  1   4   28   280    3640     58240... A007559
1  1   5   45   585    9945    208845... A007696
1  1   6   66  1056   22176    576576... A008548
1  1   7   91  1729   43225   1339975... A008542
1  1   8  120  2640   76560   2756160... A045754
1  1   9  153  3825  126225   5175225... A045755
1  1  10  190  5320  196840   9054640... A045756
1  1  11  231  7161  293601  14977651... A144773

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Cf. Columns : A000012, A000012, A000384, A011199, A011245.
Cf. Diagonals : A092985, A076111, A158887.
Cf. A048994, A132393.
Cf. A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A144773 (10)

Flat(List([0..12], n-> List([0..n], k-> Product([0..n-k-1], j-> j*k+1) ))); # G. C. Greubel, Mar 04 2020

function T(n,k)
  if k eq 0 or n eq 0 then return 1;
  else return (&*[j*k+1: j in [0..n-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 04 2020

seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # G. C. Greubel, Mar 04 2020

T[n_, k_]= Product[j*k+1, {j,0,n-1}]; Table[T[n-k,k], {n,0,12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 04 2020 *)

T(n,k) = prod(j=0, n-1, j*k+1);
for(n=0,12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 04 2020

[[ product(j*k+1 for j in (0..n-k-1)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 04 2020

A(n, k) = (-n)^k*FallingFactorial(-1/n, k) for n >= 1. - Peter Luschny, Dec 21 2021

A076110 Triangle (read by rows) in which the n-th row contains first n terms of an arithmetic progression with first term 1 and common difference (n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 7, 10, 1, 5, 9, 13, 17, 1, 6, 11, 16, 21, 26, 1, 7, 13, 19, 25, 31, 37, 1, 8, 15, 22, 29, 36, 43, 50, 1, 9, 17, 25, 33, 41, 49, 57, 65, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122

Offset: 1

Amarnath Murthy, Oct 09 2002

Leading diagonal contains n^2 + 1 (A002522).

Sum of the n-th row is (n+1)(n^2+2)/2 (A064808).

			1;
1, 2;
1, 3, 5;
1, 4, 7, 10;
1, 5, 9, 13, 17;
1, 6, 11, 16, 21, 26;
1, 7, 13, 19, 25, 31, 37; ...

Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)

Cf. A002522, A064808, A076111 (row products), A079904.

Flat(List([1..12],n->List([1..n],k->1+(n-1)*(k-1)))); # Muniru A Asiru, Dec 05 2018

/* As triangle */ [[1+(n-1)*(k-1): k in [1..n]]: n in [1.. 12]]; // Vincenzo Librandi, Dec 05 2018

T:= (n,k) -> 1+(n-1)*(k-1):for n from 1 to 10 do seq(T(n,k),k=1..n) od; # Robert Israel, Dec 04 2018

T[n_, k_] := 1 + (n-1) * (k-1); Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* Amiram Eldar, Dec 04 2018 *)

A076110(n) = L(n) with L=seq(seq(n*k+1, k = 0..n), n = 0..+inf). - Yalcin Aktar, Jul 14 2009
From Robert Israel, Dec 04 2018: (Start)
T(n,k) = 1 + (n-1)*(k-1).
G.f. as triangle: (1-x-x*y+2*x^2*y+2*x^2*y^2-3*x^3*y^2)*x*y/((1-x)^2*(1-x*y)^3).
G.f. as sequence: x/(1-x) + Sum_{m>=0} (-m*(m+1)*x^((m^2+3*m+4)/2) + (1+m*(m+1))*x^((m^2+3*m+6)/2))/(1-x)^2.
(End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

cp's OEIS Frontend

A256268 Table of k-fold factorials, read by antidiagonals.

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