A062707 -id:A062707 - cp's OEIS frontend

A104633 Triangle T(n,k) = k(k-n-1)(k-n-2)/2 read by rows, 1<=k<=n.

1, 3, 2, 6, 6, 3, 10, 12, 9, 4, 15, 20, 18, 12, 5, 21, 30, 30, 24, 15, 6, 28, 42, 45, 40, 30, 18, 7, 36, 56, 63, 60, 50, 36, 21, 8, 45, 72, 84, 84, 75, 60, 42, 24, 9, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 66, 110, 135

Offset: 1

Gary W. Adamson, Mar 18 2005

The triangle can be constructed multiplying the triangle A(n,k)=n-k+1 (if 1<=k<=n, else 0) by the triangle B(n,k) =k (if 1<=k<=n, else 0).

Swapping the two triangles of this matrix product would generate A104634.

			First few rows of the triangle:
  1;
  3,  2;
  6,  6,  3;
 10, 12,  9,  4;
 15, 20, 18, 12,  5;
 21, 30, 30, 24, 15,  6;
 28, 42, 45, 40, 30, 18,  7;
 36, 56, 63, 60, 50, 36, 21, 8;
 ...
e.g. Col. 3 = 3 * (1, 3, 6, 10, 15...) = 3, 9, 18, 30, 45...

G. C. Greubel, Rows n=1..100 of triangle, flattened
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
Joaquín Figueroa, Ivan Gonzalez, and Daniel Salinas-Arizmendi, A Novel Transfer Matrix Framework for Multiple Dirac Delta Potentials, arXiv:2503.23134 [quant-ph], 2025. See pp. 4, 9.

Cf. A062707, A158824, A104634, A001296, A000332 (row sums).

[[k*(k-n-1)*(k-n-2)/2: k in [1..n]]: n in [1..20]]; // G. C. Greubel, Aug 12 2018

A104633 := proc(n,k) k*(k-n-1)*(k-n-2)/2 ; end proc:
seq(seq(A104633(n,k),k=1..n),n=1..16) ; # R. J. Mathar, Mar 03 2011

Table[k*(k-n-1)*(k-n-2)/2, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Aug 12 2018 *)

for(n=1,20, for(k=1,n, print1(k*(k-n-1)*(k-n-2)/2, ", "))) \\ G. C. Greubel, Aug 12 2018

G.f.: x*y/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, May 22 2023

A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)(n-k+1)(n-k+2)/2, read by rows.

Original entry on oeis.org

1, 4, 1, 10, 3, 2, 20, 6, 6, 3, 35, 10, 12, 9, 4, 56, 15, 20, 18, 12, 5, 84, 21, 30, 30, 24, 15, 6, 120, 28, 42, 45, 40, 30, 18, 7, 165, 36, 56, 63, 60, 50, 36, 21, 8, 220, 45, 72, 84, 84, 75, 60, 42, 24, 9, 286, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 364, 66, 110, 135, 144, 140, 126, 105, 80, 54, 30, 11

Offset: 1

Gary W. Adamson & Roger L. Bagula, Mar 28 2009

The triangle can also be defined by multiplying the triangles A(n,k)=1 and A158823(n,k), that is, this here are the partial column sums of A158823.

			First few rows of the triangle are:
    1;
    4,  1;
   10,  3,   2;
   20,  6,   6,   3;
   35, 10,  12,   9,   4;
   56, 15,  20,  18,  12,   5;
   84, 21,  30,  30,  24,  15,   6;
  120, 28,  42,  45,  40,  30,  18,   7;
  165, 36,  56,  63,  60,  50,  36,  21,   8;
  220, 45,  72,  84,  84,  75,  60,  42,  24,  9;
  286, 55,  90, 108, 112, 105,  90,  70,  48, 27, 10;
  364, 66, 110, 135, 144, 140, 126, 105,  80, 54, 30, 11;
  455, 78, 132, 165, 180, 180, 168, 147, 120, 90, 60, 33, 12;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A062707, A104633, A158823.

Row sums: A000332.

A158824:= func< n,k | k eq 1 select Binomial(n+2,3) else (k-1)*Binomial(n-k+2,2) >; [A158824(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021

T[n_, k_]:= If[k==1, Binomial[n+2, 3], (k-1)*Binomial[n-k+2, 2]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

def A158824(n,k): return binomial(n+2,3) if k==1 else (k-1)*binomial(n-k+2,2)
flatten([[A158824(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

T(n,k) = binomial(n+2,3) if k = 1 otherwise (k-1)*binomial(n-k+2, 2).

Sum_{k=1..n} T(n, k) = binomial(n+3, 4) = A000332(n+3). - G. C. Greubel, Apr 01 2021

A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 6, 3, 2, 1, 5, 5, 1, 3, 5, 4, 4, 4, 2, 2, 1, 7, 6, 8, 4, 3, 5, 7, 9, 7, 6, 5, 4, 3, 2, 1, 8, 11, 7, 11, 1, 4, 5, 7, 9, 10, 9, 5, 7, 6, 2, 4, 3, 2, 1, 15, 10, 9, 9, 14, 6, 3, 5, 7, 9, 11, 12, 8, 18, 8, 8, 7, 6, 4, 4, 3, 2, 1, 13, 12, 11, 10, 12, 17, 1, 6, 5, 7, 9, 11, 13, 14, 13, 16, 6, 10, 9, 8, 2, 6, 5, 4, 3, 2, 1

Offset: 1

Boris Putievskiy, Aug 29 2024

A208233 presents an algorithm for generating permutations, where each generated permutation is self-inverse.

The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.

			Table begins:
    k=    1   2   3   4   5   6
  -----------------------------------
  n= 1:   1,  1,  3,  1,  5,  1, ...
  n= 2:   2,  2,  2,  3,  2,  5, ...
  n= 3:   3,  3,  1,  2,  3,  3, ...
  n= 4:   6,  5,  4,  4,  4,  4, ...
  n= 5:   5,  4,  8,  5,  1,  2, ...
  n= 6:   4,  6,  6, 11,  6,  6, ...
  n= 7:   7,  7,  7,  7, 14,  7, ...
  n= 8:   9, 11,  5,  9,  8, 17, ...
  n= 9:   8,  9,  9,  8, 12,  9, ...
  n= 10: 10, 10, 18, 10, 10, 15, ...
  n= 11: 15,  8, 11,  6, 11, 11, ...
  n= 12: 12, 12, 16, 12,  9, 13, ...
  n= 13: 13, 13, 13, 13, 13, 12, ...
  n= 14: 14, 19, 14, 23,  7, 14, ...
  n= 15: 11, 15, 15, 15, 15, 10, ...
  n= 16: 16, 17, 12, 21, 30, 16, ...
  n= 17: 20, 16, 17, 17, 17,  8, ...
  n= 18: 18, 18, 10, 19, 28, 18, ...
     ... .
In column 3, the first 3 blocks have lengths 3,6 and 9. In column 6, the first 2 blocks have lengths 6 and 12. Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  2,1;
  3,2,3;
  6,3,2,1;
  5,5,1,3,5;
  4,4,4,2,2,1;

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000194, A002024, A002260, A003991, A062707, A074294, A093178, A111651, A124625, A188568, A208233, A371355.

T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n*k+k^2]-k)/(2*k)]; R=n-k*(L-1)*L/2; P=(((-1)^Max[R,k*L+1-R]+1)*R-((-1)^Max[R,k*L+1-R]-1)*(k*L+1-R))/2; result=P+k*(L-1)*L/2]
Nmax=18; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

T(n,k) = P(n,k) + k*(L(n,k)-1)*L(n,k)/2 = P(n,k) + A062707(L(n-1),k), where L(n,k) = ceiling((sqrt(8*n*k+k^2)-k)/(2*k)), R(n,k) = n-k*(L(n,k)-1)*L(n,k)/2, P(n,k) = (((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))+1)*R(n,k)-((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))-1)*(k*L(n,k)+1-R(n,k)))/2.

T(n,1) = A188568(n). T(1,k) = A093178(k). T(n,n) = A124625(n). L(n,1) = A002024(n). L(n,2) = A000194(n). L(n,3) = A111651(n). L(n,4) = A371355(n). R(n,1) = A002260(n). R(n,2) = A074294(n).

cp's OEIS Frontend

A104633 Triangle T(n,k) = k(k-n-1)(k-n-2)/2 read by rows, 1<=k<=n.

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A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)(n-k+1)(n-k+2)/2, read by rows.

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A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

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cp's OEIS Frontend

A104633 Triangle T(n,k) = k*(k-n-1)*(k-n-2)/2 read by rows, 1<=k<=n.

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Magma

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A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)*(n-k+1)*(n-k+2)/2, read by rows.

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Magma

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A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

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A104633 Triangle T(n,k) = k(k-n-1)(k-n-2)/2 read by rows, 1<=k<=n.

A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)(n-k+1)(n-k+2)/2, read by rows.