A088699 -id:A088699 - cp's OEIS frontend

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 13, 34, 73, 1, 5, 21, 73, 209, 501, 1, 6, 31, 136, 501, 1546, 4051, 1, 7, 43, 229, 1045, 4051, 13327, 37633, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 394353, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 4596553

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array begins:
    1,    1,    1,    1,     1, ... A000012;
    1,    2,    3,    4,     5, ... A000027;
    3,    7,   13,   21,    31, ... A002061;
   13,   34,   73,  136,   229, ... A135859;
   73,  209,  501, 1045,  1961, ...
  501, 1546, 4051, 9276, 19081, ...
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 13;
  1, 4, 13, 34,  73;
  1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
Main diagonal gives A152059.
Similar table: A086885, A088699, A176120.

function t(n,k)
  if n eq 0 then return 1;
  else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
  end if; return t;
end function;
[t(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

@CachedFunction
def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )
def T(n,k): return t(k,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.
A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.
From Seiichi Manyama, Jan 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} binomial(n+k-1,j)/(n-j)!.
A(n,k) = n! * LaguerreL(n, k-1, -1). (End)

A094525 Square array of binomial transforms read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 28, 21, 6, 1, 1, 7, 31, 49, 49, 31, 7, 1, 1, 8, 43, 76, 89, 76, 43, 8, 1, 1, 9, 57, 109, 141, 141, 109, 57, 9, 1, 1, 10, 73, 148, 205, 226, 205, 148, 73, 10, 1, 11, 91, 193, 281

Offset: 0

Paul Barry, May 07 2004

Rows (and columns) are binomial transforms of [1,k,k(k-1),0,0,0,...], k>=0. One of a family of arrays that converge to A088699.

			Rows start
1,1,1,1,1,...
1,2,3,4,5,...
1,3,7,13,21,...
1,4,13,28,29,...
1,5,21,49,89,...

Cf. A094526.

Square array T(n, k) defined by T(n, k)=1+kn+k(k-1)n(n-1)/2

A094526 Square array of binomial transforms read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 34, 21, 6, 1, 1, 7, 31, 73, 73, 31, 7, 1, 1, 8, 43, 136, 185, 136, 43, 8, 1, 1, 9, 57, 229, 381, 381, 229, 57, 9, 1, 1, 10, 73, 358, 685, 826, 685, 358, 73, 10, 1, 1, 11, 91, 529, 1121, 1531

Offset: 0

Paul Barry, May 07 2004

Rows (and columns) are binomial transforms of [1,k,k(k-1),k(k-1)(k-2),0,0,...], k>=0. One of a family of arrays that converge to A088699.

			Rows start
1,1,1,1,1,...
1,2,3,4,5,...
1,3,7,13,21,...
1,4,13,34,73,...
1,5,21,73,185,...

Cf. A094525.

Square array T(n, k) defined by T(n, k)=1+kn+k(k-1)n(n-1)/2+k(k-1)(k-2)n(n-1)(n-2)/6

A329655 Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 33, 20, 5, 6, 30, 72, 72, 30, 6, 7, 42, 135, 208, 135, 42, 7, 8, 56, 228, 500, 500, 228, 56, 8, 9, 72, 357, 1044, 1545, 1044, 357, 72, 9, 10, 90, 528, 1960, 4050, 4050, 1960, 528, 90, 10, 11, 110, 747, 3392, 9275, 13326, 9275, 3392, 747, 110, 11

Offset: 1

Roy S. Freedman, Nov 18 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B. A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2. A relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

			The symmetric array T(n,k) begins:
  1,   2,    3,    4,     5,      6,       7,       8,        9, ...
  2,   6,   12,   20,    30,     42,      56,      72,       90, ...
  3,  12,   33,   72,   135,    228,     357,     528,      747, ...
  4,  20,   72,  208,   500,   1044,    1960,    3392,     5508, ...
  5,  30,  135,  500,  1545,   4050,    9275,   19080,    36045, ...
  6,  42,  228, 1044,  4050,  13326,   37632,   93288,   207774, ...
  7,  56,  357, 1960,  9275,  37632,  130921,  394352,  1047375, ...
  8,  72,  528, 3392, 19080,  93288,  394352, 1441728,  4596552, ...
  9,  90,  747, 5508, 36045, 207774, 1047375, 4596552, 17572113, ...

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

The diagonal T(n,n) is A097662. T(1,k)=A000027; T(2,k)=A002378; T(3,k)=A054602.

T:= (n,k)-> value(Sum(binomial(n,j)*binomial(k, j)*j!, j=1..k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

T[n_, k_] := Sum[Binomial[n, j] * Binomial[k, j] * j!, {j, 1, k}]; Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

T:=(n,k)->_plus (binomial(n,j)*binomial(k, j)* j! $ j=1..k):

T(n,k) = Sum_{j=1..k} binomial(n,j)*binomial(k,j)*j!.

T(n,k) = A088699(n,k)-1.

cp's OEIS Frontend

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

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A094525 Square array of binomial transforms read by antidiagonals.

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A094526 Square array of binomial transforms read by antidiagonals.

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A329655 Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique.

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