A121662 -id:A121662 - cp's OEIS frontend

A285268 Triangle read by rows: T(m,n) = Sum_{i=1..n} P(m,i) where P(m,n) = m!/(m-n)! is the number of permutations of m items taken n at a time, for 1 <= n <= m.

1, 2, 4, 3, 9, 15, 4, 16, 40, 64, 5, 25, 85, 205, 325, 6, 36, 156, 516, 1236, 1956, 7, 49, 259, 1099, 3619, 8659, 13699, 8, 64, 400, 2080, 8800, 28960, 69280, 109600, 9, 81, 585, 3609, 18729, 79209, 260649, 623529, 986409, 10, 100, 820, 5860, 36100, 187300, 792100, 2606500, 6235300, 9864100

Offset: 1

Rick Nungester, Apr 15 2017

T(m, n) is the total number of ordered sets of size 1 to n that can be created from m distinct items. For example, for 4 items taken 1 to 3 at a time, there are P(4, 1) + P(4, 2) + P(4, 3) = 4 + 12 + 24 = 40 total sets: 1, 12, 123, 124, 13, 132, 134, 14, 142, 143, 2, 21, 213, 214, 23, 231, 234, 24, 241, 243, 3, 31, 312, 314, 32, 321, 324, 34, 341, 342, 4, 41, 412, 413, 42, 421, 423, 43, 431, 432.
T(m, n) is the total number of tests in a software program that generates all P(m, n) possible solutions to a problem, allowing early-termination testing on each partial permutation, and not doing any such early termination. For example, from a deck of 52 cards, evaluate all possible 5-card deals as each card is dealt. T(52, 5) = 318507904 total evaluations.
T(m, n) counts the numbers with <= n distinct nonzero digits in base m+1. - M. F. Hasler, Oct 10 2019

			Triangle begins:
  1;
  2,  4;
  3,  9,  15;
  4, 16,  40,   64;
  5, 25,  85,  205,   325;
  6, 36, 156,  516,  1236,  1956;
  7, 49, 259, 1099,  3619,  8659,  13699;
  8, 64, 400, 2080,  8800, 28960,  69280, 109600;
  9, 81, 585, 3609, 18729, 79209, 260649, 623529, 986409;
  ...

Rick Nungester, Table of n, a(n) for n = 1..20100

Mirror of triangle A121662.
Row sums give A030297.
Diagonals (1..4): A007526 (less the initial 0), A038156 (less the initial 0, 0), A224869 (less the initial -1, 0), A079750 (less the initial 0).
Columns (1..3): A000027, A000290 (less the initial 0, 1), A053698 (less the initial 1, 4).

SumPermuteTriangle := proc(M)
  local m;
  for m from 1 to M do print(seq(add(m!/(m-k)!, k=1..n), n=1..m)) od;
end:
SumPermuteTriangle(10);
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k<1, 0, T(n-1, k-1)*n+n)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 26 2022

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j==0, 0, Floor[j! E - 1]]; T[m_,n_] = b[m] - m! b[m-n]/(m-n)!; Table[T[m, n],{m, 24},{n, m}]//Flatten
(* Manfred Boergens, Jun 22 2022 *)

A285268(m,n,s=m-n+1)={for(k=m-n+2,m,s=(s+1)*k);s} \\ Much faster than sum(k=1,n,m!\(m-k)!), e.g., factor 6 for m=1..99, factor 57 for m=1..199.
apply( A285268_row(m)=vector(m,n,A285268(m,n)), [1..9]) \\ M. F. Hasler, Oct 10 2019

T(n, k) = {exp(1)*(incgam(n+1, 1) - incgam(n-k, 1)*(n-k)*n!/(n-k)!) - 1;}
  apply(Trow(n) = vector(n, k, round(T(n, k))), [1..10]) \\ Adjust the realprecision if needed. Peter Luschny, Oct 10 2019

T(m, n) = Sum_{k=1..n} m!/(m-k)! = m*(1 + (m-1)*(1 + (m-2)*(1 + ... + m-n+1)...)), cf. PARI code. - M. F. Hasler, Oct 10 2019
T(n, k) = exp(1)*(Gamma(n+1, 1) - Gamma(n-k, 1)*(n-k)*n!/(n-k)!) - 1. - Peter Luschny, Oct 10 2019
Sum-free and Gamma-free formula: T(m, n) = b(m) - m!*b(m-n)/(m-n)! where b(0)=0, b(j)=floor(j!*e-1) for j>0. - Manfred Boergens, Jun 22 2022

A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

Original entry on oeis.org

1, 6, 4, 27, 21, 9, 124, 100, 52, 16, 645, 525, 285, 105, 25, 3906, 3186, 1746, 666, 186, 36, 27391, 22351, 12271, 4711, 1351, 301, 49, 219192, 178872, 98232, 37752, 10872, 2472, 456, 64, 1972809, 1609929, 884169, 339849, 97929, 22329, 4185, 657, 81, 19728190, 16099390, 8841790, 3398590, 979390, 223390, 41950, 6670, 910, 100

Offset: 1

Thomas Wieder, Aug 15 2006

The first column is A030297 = a(n) = n*(n+a(n-1)). The main diagonal are the squares A000290 = n^2. The first lower diagonal (6,21,52,...) is A069778 = q-factorial numbers 3!_q. See also A121662.

			Triangle begins:
      1
      6     4
     27    21     9
    124   100    52   16
    645   525   285  105  25
   3906  3186  1746  666  186  36
  27391 22351 12271 4711 1351 301 49
  ...

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Cf. A030297, A000290, A069778, A121662.

Row sums give A337001.

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, (T(i-1, j)+i)*i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

T[n_, k_] /; 1 <= k <= n := T[n, k] = (T[n-1, k]+n)*n;
T[, ] = 0;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2022 *)

def T(i, j): return (T(i-1, j)+i)*i if 1 <= j <= i else 0
print([T(r, c) for r in range(1, 11) for c in range(1, r+1)]) # Michael S. Branicky, Jun 22 2022

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

A224869 a(n) = n*( a(n-1)+1 ), initialized by a(1) = -1.

Original entry on oeis.org

-1, 0, 3, 16, 85, 516, 3619, 28960, 260649, 2606500, 28671511, 344058144, 4472755885, 62618582404, 939278736075, 15028459777216, 255483816212689, 4598708691828420, 87375465144739999, 1747509302894800000, 36697695360790800021, 807349297937397600484

Offset: 1

Alex Ratushnyak, Jul 22 2013

			a(4) = 4*(a(3)+1) = 4*4 = 16.

Cf. A007526, A038156, A166554, A121662 (3rd column).

A224869[1] := -1; A224869[n_] := A224869[n - 1]n + n; Table[A224869[n], {n, 40}] (* Alonso del Arte, Jul 22 2013 *)

a=-1
for n in range(2,24):
  print(a, end=', ')
  a= n*(a+1)

a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Jul 28 2013

a(n) = exp(1)*n*Gamma(n,1) - 2*Gamma(n+1,0). - Martin Clever, Mar 27 2023

A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601, 1, 10, 82, 586, 3610, 18730, 79210, 260650, 623530, 986410

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

			Triangle begins:
  1;
  1, 2;
  1, 3,  5;
  1, 4, 10,  16;
  1, 5, 17,  41,   65;
  1, 6, 26,  86,  206,  326;
  1, 7, 37, 157,  517, 1237, 1957;
  1, 8, 50, 260, 1100, 3620, 8660, 13700;
  ...

T(n,n) = A000522, T(2*n,n) = A066211.

Cf. A008279, A008949, A111063, A121662, A285268.

T[n_, k_] := Sum[Binomial[n, j] j!, {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

cp's OEIS Frontend

A285268 Triangle read by rows: T(m,n) = Sum_{i=1..n} P(m,i) where P(m,n) = m!/(m-n)! is the number of permutations of m items taken n at a time, for 1 <= n <= m.

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A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

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A224869 a(n) = n*( a(n-1)+1 ), initialized by a(1) = -1.

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A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).

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Mathematica