A056536 -id:A056536 - cp's OEIS frontend

A056537 Mapping from the column-by-column reading to the half-antidiagonal reading of the triangular tables. Inverse of A056536.

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 11, 15, 20, 25, 10, 14, 19, 24, 30, 36, 13, 18, 23, 29, 35, 42, 49, 17, 22, 28, 34, 41, 48, 56, 64, 21, 27, 33, 40, 47, 55, 63, 72, 81, 26, 32, 39, 46, 54, 62, 71, 80, 90, 100, 31, 38, 45, 53, 61, 70, 79, 89, 99, 110, 121, 37, 44, 52, 60, 69

Offset: 1

Antti Karttunen, Jun 20 2000

Moves triangular numbers (A000217) to squares (A000290), i.e., A056537(A000217(i)) = A000290(i) for i >= 1.

As a square array, this is the dispersion of the complement of the squares; see A082152. - Clark Kimberling, Apr 05 2003

			As a square array, a northwest corner:
1 ... 2 ... 3 ... 5 ... 7 ... 10
4 ... 6 ... 8 ... 11 .. 14 .. 18
9 ... 12 .. 15 .. 19 .. 23 .. 28
16 .. 20 .. 24 .. 29 .. 34 .. 40
25 .. 30 .. 35 .. 41 .. 47 .. 54
36 .. 42 .. 48 .. 55 .. 62 .. 70
49 .. 56 .. 63 .. 71 .. 79 .. 88
64 .. 72 .. 80 .. 89 .. 98 .. 108
- _Clark Kimberling_, Aug 08 2013

Cf. A185787 (dispersion of complement of triangular numbers).

Cf. A082152 (dispersion of complement of pentagonal numbers).

# using Maple procedure nthmember given in A054426:
[seq(nthmember(j, A056536), j=1..105)];

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := n+Floor[1/2+Sqrt[n]] (* complement of column 1 *); mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A056537 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A056537 sequence *)
(* Clark Kimberling, Jun 06 2011 *)

Triangle T(n, k), 1<=k<=n, read by rows, defined by: T(n, k) = 0 for nA002620(n-k+1) + k*n + k - n if n>=k. T(n, n) = n^2; T(n, 1) = 1 + A002620(n) = A033638(n). - Philippe Deléham, Feb 16 2004

Square: t(n,k) = (n-1)(n+k) + k^2/4 + (1/8)(7+(-1)^k). - Clark Kimberling, Aug 08 2013

A082156 Dispersion of the complement of row 1 of A056536.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 11, 7, 36, 30, 24, 19, 14, 10, 49, 42, 35, 29, 23, 18, 13, 64, 56, 48, 41, 34, 28, 22, 17, 81, 72, 63, 55, 47, 40, 33, 27, 21, 100, 90, 80, 71, 62, 54, 46, 39, 32, 26, 121, 110, 99, 89, 79, 70, 61, 53, 45, 38, 31, 144, 132, 120, 109

Offset: 1

Clark Kimberling, Apr 05 2003

Rectangular array read by antidiagonals; a permutation of the natural numbers. (Row 1) = squares = A000290(n) = n^2. (Dispersion of complement of column 1 of A082156) = (Transpose of A056537). The associated fractal sequence is A122196.

			Northwest corner:
1 4 9 16 25
2 6 12 20 30
3 8 15 24 35
5 11 19 29 41
7 14 23 34 47

C. Kimberling, Interspersions and Dispersions.
C. Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A056536, A000290.

The transpose of A056536.

A008315 Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 5, 1, 6, 14, 14, 1, 7, 20, 28, 14, 1, 8, 27, 48, 42, 1, 9, 35, 75, 90, 42, 1, 10, 44, 110, 165, 132, 1, 11, 54, 154, 275, 297, 132, 1, 12, 65, 208, 429, 572, 429, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 15, 104

Offset: 0

N. J. A. Sloane

There are several versions of a Catalan triangle: see A009766, A008315, A028364, A053121.
Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: T(4,1)=3 because in th top row we can have 124, 134, or 123 (but not 234). - Emeric Deutsch, May 23 2004
T(n,k) is the number of n-digit binary words (length n sequences on {0,1}) containing k 1's such that no initial segment of the sequence has more 1's than 0's. - Geoffrey Critzer, Jul 31 2009
T(n,k) is the number of dispersed Dyck paths (i.e. Motzkin paths with no (1,0) steps at positive heights) of length n and having k (1,1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), H=(1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, May 30 2011
T(n,k) is the number of length n left factors of Dyck paths having k (1,-1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), we have UUUUD, UUUDU, UUDUU, and UDUUU. There is a simple bijection between length n left factors of Dyck paths and dispersed Dyck paths of length n, that takes D steps into D steps. - Emeric Deutsch, Jun 19 2011
Triangle, with zeros omitted, given by (1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...) DELTA (0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
T(n,k) are rational multiples of A055151(n,k). - Peter Luschny, Oct 16 2015
T(2*n,n) = Sum_{k>=0} T(n,k)^2 = A000108(n), T(2*n+1,n) = A000108(n+1). - Michael Somos, Jun 08 2020

			Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 3,  2;
  1, 4,  5;
  1, 5,  9,  5;
  1, 6, 14, 14;
  1, 7, 20, 28, 14;
  ...
T(5,2) = 5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0}. - _Geoffrey Critzer_, Jul 31 2009

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.

T. D. Noe, Rows n=0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.
Nantel Bergeron, Kelvin Chan, Yohana Solomon, Farhad Soltani, and Mike Zabrocki, Quasisymmetric harmonics of the exterior algebra, arXiv:2206.02065 [math.CO], 2022.
Chassidy Bozeman, Christine Cheng, Pamela E. Harris, Stephen Lasinis, and Shanise Walker, The Pinnacle Sets of a Graph, arXiv:2406.19562 [math.CO], 2024. See pp. 9-10.
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
Suyuong Choi and Younghan Yoon, A decomposition of graph a-numbers, arXiv:2508.06855 [math.CO], 2025. See p. 14.
C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Lin Jiu, Victor H. Moll, and C. Vignat, Identities for generalized Euler polynomials, arXiv:1401.8037 [math.PR], 2014.
Nik Lygeros and Oliver Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Mustafa A. A. Obaid, S. Khalid Nauman, Wafaa M. Fakieh, and Claus Michael Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6.
Alon Regev, The central component of a triangulation, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.1, p. 7.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]
Zheng Shi, Impurity entropy of junctions of multiple quantum wires, arXiv preprint arXiv:1602.00068 [cond-mat.str-el], 2016 (See Appendix A).
Index entries for sequences related to Chebyshev polynomials.

T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial coefficients).
This is also the positive half of the triangle in A008482. - Michael Somos
This is another reading (by shallow diagonals) of the triangle A009766, i.e. A008315[n] = A009766[A056536[n]].
Cf. A120730, A055151.

a008315 n k = a008315_tabf !! n !! k
a008315_row n = a008315_tabf !! n
a008315_tabf = map reverse a008313_tabf
-- Reinhard Zumkeller, Nov 14 2013

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, n-2*k):
seq(seq(T(n, k), k=0..n/2), n=0..16);  # Alois P. Heinz, Oct 14 2022

Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/2]}] // Grid (* Geoffrey Critzer, Jul 31 2009 *)

{T(n, k) = if( k<0 || k>n\2, 0, if( n==0, 1, T(n-1, k-1) + T(n-1, k)))}; /* Michael Somos, Aug 17 1999 */

T(n, 0) = 1 if n >= 0; T(2*k, k) = T(2*k-1, k-1) if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) if k=1, 2, ..., floor(n/2). - Michael Somos, Aug 17 1999
T(n, k) = binomial(n, k) - binomial(n, k-1). - Michael Somos, Aug 17 1999
Rows of Catalan triangle A008313 read backwards. Sum_{k>=0} T(n, k)^2 = A000108(n); A000108 : Catalan numbers. - Philippe Deléham, Feb 15 2004
T(n,k) = C(n,k)*(n-2*k+1)/(n-k+1). - Geoffrey Critzer, Jul 31 2009
Sum_{k=0..n} T(n,k)*x^k = A000012(n), A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 12 2011

Expanded description from Clark Kimberling, Jun 15 1997

A079823 Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.

Original entry on oeis.org

1, 2, 43, 75, 1186, 16129, 22171310, 29231814, 3730241915, 4638312520, 564739322621, 675748403327, 79685849413428, 92806959504235, 10693817060514336, 121107948271615244, 137122108958372625345, 1541381231099684736354, 1721551391241109785746455

Offset: 1

Amarnath Murthy, Feb 11 2003

   1
   2  3
   4  5  6
   7  8  9 10
  11 12 13 14 15
  16 17 18 19 20 21
  ...
a(n) also is the concatenation of the terms of the n-th row of A056536. - Michel Marcus, Dec 14 2023

G. C. Greubel, Table of n, a(n) for n = 1..395

Cf. A056536, A079808, A079824.

read("transforms"):
A079823aux := proc(n,k)
    A000124(n)+k ;
end proc:
A079823 := proc(n)
    local L,k,n0 ;
    n0 := n-1 ;
    L := [] ;
    for k from 0 do
        if k > n0-k then
            break;
        end if;
        L := [op(L),A079823aux(n0-k,k)] ;
    end do:
    digcatL(L) ;
end proc: # R. J. Mathar, Aug 23 2012
# second Maple program:
T:= (i, j)-> i*(i-1)/2+j:
a:= n-> parse(cat(seq(T(n-j,j+1), j=0..(n-1)/2))):
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 03 2022

Table[FromDigits[Join@@IntegerDigits[Table[Binomial[n-k+1,2] + k, {k, Ceiling[n/2]}]]], {n,30}] (* G. C. Greubel, Dec 13 2023 *)

More terms from Jason D. W. Taff (jtaff(AT)jburroughs.org), Oct 31 2003

Corrected by Philippe Deléham, Feb 16 2004

A091018 Numbers in n-th upward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; ...

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 10, 7, 5, 15, 11, 8, 21, 16, 12, 9, 28, 22, 17, 13, 36, 29, 23, 18, 14, 45, 37, 30, 24, 19, 55, 46, 38, 31, 25, 20, 66, 56, 47, 39, 32, 26, 78, 67, 57, 48, 40, 33, 27, 91, 79, 68, 58, 49, 41, 34, 105, 92, 80, 69, 59, 50, 42, 35, 120, 106, 93, 81, 70, 60, 51

Offset: 0

Philippe Deléham, Feb 25 2004

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

cp's OEIS Frontend

A056537 Mapping from the column-by-column reading to the half-antidiagonal reading of the triangular tables. Inverse of A056536.

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A082156 Dispersion of the complement of row 1 of A056536.

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A008315 Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

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A079823 Consider the triangle shown below; sequence contains the concatenation of numbers read at a 45-degree angle upwards with horizontal beginning with the first term of a row.

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A091018 Numbers in n-th upward diagonal of triangle T : 0; 1, 2; 3, 4, 5; 6, 7, 8, 9; ...

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A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

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