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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A371766 Triangle read by rows: T(n, k) = A371898(n, k) / A371767(n, k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 16, 21, 7, 1, 1, 65, 142, 63, 11, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1
Offset: 0

Views

Author

Peter Luschny, Apr 14 2024

Keywords

Examples

			Triangle starts:
  [0] 1;
  [1] 1,     1;
  [2] 1,     2,      1;
  [3] 1,     5,      4,      1;
  [4] 1,    16,     21,      7,     1;
  [5] 1,    65,    142,     63,    11,    1;
  [6] 1,   326,   1201,    709,   151,   16,   1;
  [7] 1,  1957,  12336,   9709,  2521,  311,  22,  1;
  [8] 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1;

Crossrefs

Antidiagonally read subtriangle of A181783.
Cf. A371898, A371767.

Programs

  • Maple
    A371766 := (n, k) -> local j; add((-1)^(k-j)*binomial(k, j)*hypergeom([1, -n],
[], -j), j = 0..k)/((k! * n!)/(n - k)!):
seq(print(seq(simplify(A371766(n, k)), k = 0..n)), n = 0..8);

A181783 Array described in comments to A053482, here read by increasing antidiagonals. See comments below.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 16, 21, 7, 1, 1, 1, 65, 142, 63, 11, 1, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1
Offset: 0

Views

Author

Richard Choulet, Dec 23 2012

Keywords

Comments

We denote by a(n,k) the number in row number n >= 0 and column number k >= 0. The recurrence which defines the array is a(n,k) = n*(k-1)*a(n-1,k) + a(n,k-1). The initial values are given by a(n,0) = 1 = a(0,k) for all n >= 0 and k >= 0.

Examples

			Array read row after row:
  1, 1,    1,      1,       1,        1,         1, ...
  1, 1,    2,      4,       7,       11,        16, ...
  1, 1,    5,     21,      63,      151,       311, ...
  1, 1,   16,    142,     709,     2521,      7186, ...
  1, 1,   65,   1201,    9709,    50045,    193765, ...
  1, 1,  326,  12336,  157971,  1158871,   6002996, ...
  1, 1, 1957, 149989, 2993467, 30806371, 210896251, ...
  ...
A(4,3) = 1201.

Crossrefs

Cf. A000522, A053482, A185106, A371898.

Programs

  • Maple
    A181783 := proc(n,k)
    option remember;
    if n =0 or k = 0 then
        1;
    else
        n*(k-1)*procname(n-1,k)+procname(n,k-1) ;
    end if;
end proc:
seq(seq(A181783(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Mar 02 2016
  • Mathematica
    T[n_, k_] := T[n, k] = If[n == 0 || k == 0, 1, n (k - 1) T[n - 1, k] + T[n, k - 1]];
Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2023 *)

Formula

If we consider the e.g.f. Psi(k) of column number k we have: Psi(k)(z) = Psi(k-1)(z)/(1-(k-1)*z) with Psi(1)(z) = exp(z). Then Psi(k)(z) = exp(z)/Product_{j=0..k-1} (1 - j*z). We conclude that a(n,k) = n!*Sum_{m=0..n} Sum_{j=1..k-1} (-1)^(k-1-j)*j^(m+k-2)/((n-m)!*(j-1)!*(k-1-j)!). It seems after the recurrence (and its proof) in A053482 that:
A(n,k) = -Sum_{j=1..k-1} s1(k,k-j)*n*(n-1)*...*(n-k+1)*a(n-j,k) + 1 where s1(m,n) are the classical Stirling numbers of the first kind.
A(n,1) = 1 for every n.
A(1,k) = 1 + k*(k-1)/2 for every k.
A(n, k+1) = A371898(n+k, k) * n! / ((n+k)! * k!). - Werner Schulte, Apr 14 2024

Extensions

Edited by N. J. A. Sloane, Dec 24 2012

A320031 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(x)/(1 - k*x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 16, 1, 1, 5, 25, 79, 65, 1, 1, 6, 41, 226, 633, 326, 1, 1, 7, 61, 493, 2713, 6331, 1957, 1, 1, 8, 85, 916, 7889, 40696, 75973, 13700, 1, 1, 9, 113, 1531, 18321, 157781, 732529, 1063623, 109601, 1, 1, 10, 145, 2374, 36745, 458026, 3786745, 15383110, 17017969, 986410, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2018

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (2*k^2 + 2*k + 1)*x^2/2! + (6*k^3 + 6*k^2 + 3*k + 1)*x^3/3! + (24*k^4 + 24*k^3 + 12*k^2 + 4*k + 1)*x^4/4! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  1,    2,     3,      4,       5,       6,  ...
  1,    5,    13,     25,      41,      61,  ...
  1,   16,    79,    226,     493,     916,  ...
  1,   65,   633,   2713,    7889,   18321,  ...
  1,  326,  6331,  40696,  157781,  458026,  ...

Links

Crossrefs

Columns k=0..6 give A000012, A000522, A010844, A010845, A056545, A056546, A056547.
Main diagonal gives A277452.
Cf. A007318, A320032, A371898.

Programs

  • Maple
    A := (n, k) -> simplify(hypergeom([1, -n], [], -k)):
for n from 0 to 5 do seq(A(n, k), k=0..8) od; # Peter Luschny, Oct 03 2018
# second Maple program:
A:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*A(n-1, k), 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 09 2020
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, HypergeometricPFQ[{1, -n}, {}, -k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(x)/(1 - k*x).
A(n,k) = Sum_{j=0..n} binomial(n,j)*j!*k^j.
A(n,k) = hypergeom_2F0([1, -n], [], -k).
A(n,k) = 1 + [n > 0] * k * n * A(n-1,k). - Alois P. Heinz, May 09 2020
A(n,k) = floor(n!*k^n*exp(1/k)), k > 0, n + k > 1. - Peter McNair, Dec 20 2021
From Werner Schulte, Apr 14 2024: (Start)
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371898, i.e., A(n, k) = Sum_{i=0..k} binomial(k, i) * A371898(n, i).
Conjecture: E.g.f. of row n is exp(x) * (Sum_{k=0..n} A371898(n, k) * x^k / k!). (End)
Showing 1-3 of 3 results.

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