A054124 -id:A054124 - cp's OEIS frontend

A054125 Sum of the arrays in A054123 and A054124.

2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 4, 4, 4, 2, 2, 5, 6, 6, 5, 2, 2, 6, 9, 8, 9, 6, 2, 2, 7, 13, 12, 12, 13, 7, 2, 2, 8, 18, 19, 16, 19, 18, 8, 2, 2, 9, 24, 30, 24, 24, 30, 24, 9, 2, 2, 10, 31, 46, 39, 32, 39, 46, 31, 10, 2, 2, 11, 39, 68, 65, 48, 48, 65

Offset: 0

Clark Kimberling

Row sums are twice Fibonacci numbers, A006355(n+2).

			Rows:
  2;
  2,2;
  2,2,2;
  2,3,3,2;
  ...

A052509(n,k) = sum(m=0, k, binomial(n-k, m));
T(n,k) = if(k==0 || k==n, 2, A052509(n-1,k) + A052509(n-1,n-k)) \\ Jianing Song, May 30 2022

From Jianing Song, May 30 2022: (Start)
T(n,k) = 2 if k = 0 or k = n, A052509(n-1,k) + A052509(n-1,n-k) otherwise.
G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) * (1/((1-x*y)*(1-x-x^2*y)) + 1/((1-x)*(1-x*y-x^2*y))). (End)

A052509 Knights-move Pascal triangle: T(n,k), n >= 0, 0 <= k <= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 11, 8, 4, 2, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 8, 22, 26, 16, 8, 4, 2, 1, 1, 9, 29, 42, 31, 16, 8, 4, 2, 1, 1, 10, 37, 64, 57, 32, 16, 8, 4, 2, 1, 1, 11, 46, 93, 99, 63, 32, 16, 8, 4, 2, 1

Offset: 0

N. J. A. Sloane, Mar 17 2000

Also square array T(n,k) (n >= 0, k >= 0) read by antidiagonals: T(n,k) = Sum_{i=0..k} binomial(n,i).
As a number triangle read by rows, this is T(n,k) = Sum_{i=n-2*k..n-k} binomial(n-k,i), with T(n,k) = T(n-1,k) + T(n-2,k-1). Row sums are A000071(n+2). Diagonal sums are A023435(n+1). It is the reverse of the Whitney triangle A004070. - Paul Barry, Sep 04 2005
Also, twice number of orthants intersected by a generic k-dimensional subspace of R^n [Naiman and Scheinerman, 2017]. - N. J. A. Sloane, Mar 03 2018

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  2,  1;
[4] 1, 4,  4,  2,  1;
[5] 1, 5,  7,  4,  2,  1;
[6] 1, 6, 11,  8,  4,  2, 1;
[7] 1, 7, 16, 15,  8,  4, 2, 1;
[8] 1, 8, 22, 26, 16,  8, 4, 2, 1;
[9] 1, 9, 29, 42, 31, 16, 8, 4, 2, 1;
As a square array, this begins:
  1  1  1  1  1  1 ...
  1  2  2  2  2  2 ...
  1  3  4  4  4  4 ...
  1  4  7  8  8  8 ...
  1  5 11 15 16 ...
  1  6 16 26 31 32 ...

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1C.
Daniel Q. Naiman and Edward R. Scheinerman, Arbitrage and Geometry, arXiv:1709.07446 [q-fin.MF], 2017 [Contains the square array multiplied by 2].
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Tables 5 and 14.
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A054123, A054124, A007318, A008949.
Row sums A000071; Diagonal sums A023435; Mirror A004070.
Columns give A000027, A000124, A000125, A000127, A006261, ...
Cf. A052509, A054123, A054124, A007318, A008949, A052553.
Partial sums across rows of (extended) Pascal's triangle A052553.

A052509:=Flat(List([0..100],n->List([0..n],k->Sum([0..n],m->Binomial(n-k,k-m))))); # Muniru A Asiru, Sat Feb 17 2018

a052509 n k = a052509_tabl !! n !! k
a052509_row n = a052509_tabl !! n
a052509_tabl = [1] : [1,1] : f [1] [1,1] where
   f row' row = rs : f row rs where
     rs = zipWith (+) ([0] ++ row' ++ [1]) (row ++ [0])
-- Reinhard Zumkeller, Nov 22 2012

[[(&+[Binomial(n-k, k-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 13 2019

a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) fi: if k=n then RETURN(1) fi: a(n-1,k)+a(n-2,k-1) end: for n from 0 to 11 do for k from 0 to n do printf(`%d,`,a(n,k)) od: od: # James Sellers, Mar 17 2000
with(combinat): for s from 0 to 11 do for n from s to 0 by -1 do if n=0 or s-n=0 then printf(`%d,`,1) else printf(`%d,`,sum(binomial(n, i), i=0..s-n)) fi; od: od: # James Sellers, Mar 17 2000

Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}]
T[n_, k_] := Hypergeometric2F1[-k, -n + k, -k, -1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 28 2021 *)

T(n,k)=sum(m=0,n,binomial(n-k,k-m));
for(n=0,10,for(k=0,n,print1(T(n,k),", "););print();); /* show triangle */

[[sum(binomial(n-k, k-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 13 2019

T(n, k) = Sum_{m=0..n} binomial(n-k, k-m). - Wouter Meeussen, Oct 03 2002
From Werner Schulte, Feb 15 2018: (Start)
Referring to the square array T(i,j):
G.f. of row n: Sum_{k>=0} T(n,k) * x^k = (1+x)^n / (1-x).
G.f. of T(i,j): Sum_{k>=0, n>=0} T(n,k) * x^k * y^n = 1 / ((1-x)*(1-y-x*y)).
Let a_i(n) be multiplicative with a_i(p^e) = T(i, e), p prime and e >= 0, then Sum_{n>0} a_i(n)/n^s = (zeta(s))^(i+1) / (zeta(2*s))^i for i >= 0.
(End)
T(n, k) = hypergeom([-k, -n + k], [-k], -1). - Peter Luschny, Nov 28 2021
From Jianing Song, May 30 2022: (Start)
Referring to the triangle, G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = 1 / ((1-x*y)*(1-x-x^2*y)).
T(n,k) = 2^(n-k) for ceiling(n/2) <= k <= n. (End)

More terms from James Sellers, Mar 17 2000
Entry formed by merging two earlier entries. - N. J. A. Sloane, Jun 17 2007
Edited by Johannes W. Meijer, Jul 24 2011

A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 0, 1, 12, 55

Offset: 0

N. J. A. Sloane, Mar 17 2000

Another version of Pascal's triangle A007318.
As a triangle read by rows, it is (1,0,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938 and it is the Riordan array (1/(1-x), x^2/(1-x)). The row sums of this triangle are F(n+1) = A000045(n+1). - Philippe Deléham, Dec 11 2011
As a triangle, binomial(n-k, k) is also the number of ways to add k pierced circles to a path graph P_n so that no two circles share a vertex (see Lemma 3.1 at page 5 in Owad and Tsvietkova). - Stefano Spezia, May 18 2022
For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023

			Array begins:
  1, 0,  0,  0, 0, 0, ...
  1, 1,  0,  0, 0, 0, ...
  1, 2,  1,  0, 0, 0, ...
  1, 3,  3,  1, 0, 0, ...
  1, 4,  6,  4, 1, 0, ...
  1, 5, 10, 10, 5, 1, ...
As a triangle, this begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  0, 0;
  1, 3,  1, 0, 0;
  1, 4,  3, 0, 0, 0;
  1, 5,  6, 1, 0, 0, 0;
  1, 6, 10, 4, 0, 0, 0, 0;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5459
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.
Index entries for triangles and arrays related to Pascal's triangle

The official entry for Pascal's triangle is A007318. See also A026729 (the same array read by downward antidiagonals).
As a triangle without zeros: A011973.
Cf. A052509, A054123, A054124, A008949.

/* As triangle */ [[Binomial(n-k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 08 2017

with(combinat): for s from 0 to 20 do for n from s to 0 by -1 do printf(`%d,`, binomial(n, s-n)) od:od: # James Sellers, Mar 17 2000

Flatten[ Table[ Binomial[n-k , k], {n, 0, 13}, {k, 0, n}]]  (* Jean-François Alcover, Dec 05 2012 *)

T(n,k) = binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017

As a triangle: T(n,k) = A026729(n,n-k).

G.f. of the triangular version: 1/(1-x-x^2*y). - R. J. Mathar, Aug 11 2015

A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 1, 1, 1, 5, 7, 4, 2, 1, 1, 1, 6, 11, 8, 4, 2, 1, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 1, 8, 22, 26, 16, 8, 4, 2, 1, 1, 1, 9, 29, 42, 31, 16, 8, 4, 2, 1, 1, 1, 10, 37, 64, 57, 32, 16, 8, 4, 2, 1, 1, 1, 11, 46, 93, 99, 63, 32, 16, 8, 4, 2, 1, 1, 1, 12, 56

Offset: 0

Clark Kimberling

Variant of A052509 with an additional diagonal of 1's. - R. J. Mathar, Oct 12 2011
Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
  (1)  if x is in g(n-1), then x + 1 is in g(n); and
  (2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
Then g(1) = {1/1}, g(2) = {1/2,2/1}, g(3) = {1/4,3/2,3/1}, etc. The denominators in g(n) are 2^0, 2^1, ..., 2^(n-1), and T(n,k) is the number of occurrences of 2^(n-1-k), for k = 0..n-1. - Clark Kimberling, Nov 09 2015
G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) / ((1-x*y)*(1-x-x^2*y)). - Jianing Song, May 30 2022

			Rows:
1
1 1
1 1 1
1 2 1 1
1 3 2 1 1
1 4 4 2 1 1
1 5 7 4 2 1 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
S. Sivasubramanian, Signed excedance enumeration in the hyperoctahedral group El. J. Combinat. 21 (1) (2014) #P2.10, Remark 16.
Index entries for triangles and arrays related to Pascal's triangle

Reflection of array in A054124 about vertical central line.
Row sums: 1, 2, 3, 5, 8, 13, ... (Fibonacci numbers, A000045). Central numbers: 1, 1, 2, 4, 8, ... (binary powers, A000079). Cf. A011973.
Cf. A129713.

a054123 n k = a054123_tabl !! n !! k
a054123_row n = a054123_tabl !! n
a054123_tabl = [1] : [1, 1] : f [1] [1, 1] where
   f us vs = ws : f vs ws where
             ws = zipWith (+) (0 : init us ++ [0, 0]) (vs ++ [1])
-- Reinhard Zumkeller, May 26 2015

Clear[t]; t[n_, k_] := t[n, k] = If[k == 0 || k == n || k == n-1, 1, t[n-1, k] + t[n-2, k-1]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 01 2013 *)

A052509(n,k) = sum(m=0, k, binomial(n-k, m));
T(n,k) = if(k==n, 1, A052509(n-1,k)) \\ Jianing Song, May 30 2022

T(n, 0) = T(n, n) = 1 for n >= 0; T(n, n-1) = 1 for n >= 1; T(n, k) = T(n-1, k) + T(n-2, k-1) for k=1, 2, ..., n-2, n >= 3. [Corrected by Jianing Song, May 30 2022]

T(n, k) = T(n-1, k-1) + U(n-1, k) for k=1, 2, ..., floor(n/2), n >= 3, array U as in A011973.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003

A264200 Numerator of sum of numbers in set g(n) generated as in Comments.

Original entry on oeis.org

0, 1, 5, 19, 69, 235, 789, 2603, 8533, 27819, 90453, 293547, 951637, 3082923, 9983317, 32320171, 104617301, 338602667, 1095849301, 3546458795, 11477013845, 37141260971, 120193373525, 388957383339, 1258699445589, 4073250794155, 13181344109909, 42655780874923

Offset: 0

Clark Kimberling, Nov 09 2015

Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
(1)  if x is in g(n-1), then x + 1 is in g(n); and
(2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
The sum of numbers in g(n) is a(n)/2^(n-1).

			g(0) = {0}, sum = 0.
g(1) = {1}, sum = 1.
g(2) = {1/2,2/1}, sum = 5/4.
g(3) = {1/4,3/2,3/1}, sum = 19/8.

Cf. A054123, A054124, A264201.

z = 30; x = 1/2; g[0] = {0}; g[1] = {1};
g[n_] := g[n] = Union[1 + g[n - 1], (1/2) Select[g[n - 1], # < 2 &]]
Table[g[n], {n, 0, z}]; Table[Total[g[n]], {n, 0, z}]
Numerator[Table[Total[g[n]], {n, 0, z}] ]

Conjecture: a(n) = 3*a(n-1) + 4*a(n-2) - 8*a(n-3) - 8*a(n-4).

cp's OEIS Frontend

A054125 Sum of the arrays in A054123 and A054124.

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A052509 Knights-move Pascal triangle: T(n,k), n >= 0, 0 <= k <= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, n >= 2.

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A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

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A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.

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A264200 Numerator of sum of numbers in set g(n) generated as in Comments.

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