A027660 -id:A027660 - cp's OEIS frontend

A080851 Square array of pyramidal numbers, read by antidiagonals.

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78

Offset: 0

Paul Barry, Feb 21 2003

The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

			Array begins (n>=0, k>=0):
1,  3,  6, 10,  15,  21,  28,  36,  45,   55, ... A000217
1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145, A027660 (antidiagonal sums).
See A257199 for another version of this array.

vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4,x,11),x,n),n,0,11),k,-1,10) VECTOR(VECTOR(comb(k+2,2)+comb(k+2,3)n, k, 0, 11), n, 0, 11)

A080851 := proc(n,k)
    binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
end proc:
seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)

T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016

A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 15, 15, 11, 5, 6, 16, 26, 30, 26, 16, 6, 7, 22, 42, 56, 56, 42, 22, 7, 8, 29, 64, 98, 112, 98, 64, 29, 8, 9, 37, 93, 162, 210, 210, 162, 93, 37, 9, 10, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10

Offset: 0

Asher Auel

The number of spotlight tilings of an m X n rectangle missing the southeast corner. E.g., there are 2 spotlight tilings of a 2 X 2 square missing its southeast corner. - Bridget Tenner, Nov 10 2007
T(n,k) = A134636(n,k) - A051597(n,k). - Reinhard Zumkeller, Nov 23 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			From _Roger L. Bagula_, Feb 17 2009: (Start)
Triangle begins:
   0;
   1,  1;
   2,  2,   2;
   3,  4,   4,   3;
   4,  7,   8,   7,    4;
   5, 11,  15,  15,   11,    5;
   6, 16,  26,  30,   26,   16,   6;
   7, 22,  42,  56,   56,   42,   22,    7;
   8, 29,  64,  98,  112,   98,   64,   29,   8;
   9, 37,  93, 162,  210,  210,  162,   93,   37,   9;
  10, 46, 130, 255,  372,  420,  372,  255,  130,  46,  10;
  11, 56, 176, 385,  627,  792,  792,  627,  385, 176,  56, 11;
  12, 67, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 67, 12. ... (End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
B. E. Tenner, Spotlight tiling, Ann. Combinat. 14 (4) (2010) 553-568.
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A000918(n+1).
Cf. A007318, A224791, A228196, A228576.
Columns from 2 to 9, respectively: A000124; A000125, A055795, A027660, A055796, A055797, A055798, A055799 (except 1 for the last seven). [Bruno Berselli, Aug 02 2013]
Cf. A001477, A162551 (central terms).

Flat(List([0..12], n-> List([0..n], k->  Binomial(n, k+1) + Binomial(n, n-k+1) ))); # G. C. Greubel, Nov 12 2019

a051601 n k = a051601_tabl !! n !! k
a051601_row n = a051601_tabl !! n
a051601_tabl = iterate
               (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [0]
-- Reinhard Zumkeller, Nov 23 2012

/* As triangle: */ [[Binomial(n,m+1)+Binomial(n,n-m+1): m in [0..n]]: n in [0..12]]; // Bruno Berselli, Aug 02 2013

seq(seq(binomial(n,k+1) + binomial(n, n-k+1), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

T[n_, k_]:= T[n, k] = Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 17 2009; modified by G. C. Greubel, Nov 12 2019 *)

T(n,k) = binomial(n, k+1) + binomial(n, n-k+1);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 12 2019

[[binomial(n, k+1) + binomial(n, n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(m,n) = binomial(m+n,m) - 2*binomial(m+n-2,m-1), up to offset and transformation of array to triangular indices. - Bridget Tenner, Nov 10 2007
T(n,k) = binomial(n, k+1) + binomial(n, n-k+1). - Roger L. Bagula, Feb 17 2009
T(0,n) = T(n,0) = n, T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n.

A055794 Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=1 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j) = T(i-1,j) + T(i-2,j-1) for 1<=j<=i-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 0, 1, 5, 7, 4, 1, 0, 1, 6, 11, 8, 3, 0, 0, 1, 7, 16, 15, 7, 1, 0, 0, 1, 8, 22, 26, 15, 4, 0, 0, 0, 1, 9, 29, 42, 30, 11, 1, 0, 0, 0, 1, 10, 37, 64, 56, 26, 5, 0, 0, 0, 0, 1, 11, 46, 93, 98, 56, 16, 1, 0, 0, 0, 0, 1, 12, 56, 130, 162, 112, 42, 6, 0, 0, 0, 0, 0

Offset: 0

Clark Kimberling, May 28 2000

T(i+j,j) is the number of strings (s(1),...,s(i+1)) of nonnegative integers s(k) such that 0<=s(k)-s(k-1)<=1 for k=2,3,...,i+1 and s(i+1)=j.

T(i+j,j) is the number of compositions of j consisting of i parts, all of in {0,1}.

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 2, 1;
  1, 4, 4, 2, 0;
  1, 5, 7, 4, 1, 0;
  ...
T(7,4) counts the strings 3334, 3344, 3444, 2234, 2334, 2344, 1234.
T(7,4) counts the compositions 001, 010, 100, 011, 101, 110, 111.

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1B.

Row sums: A000204 (Lucas numbers).

Cf. subsequences T(2n+m,n): A000125 (m=0, cake numbers), A055795 (m=1), A027660 (m=2), A055796 (m=3), A055797 (m=4), A055798 (m=5), A055799 (m=6).

T:= function(n,k)
    if k=0 then return 1;
    elif k=n and n<4 then return 1;
    elif k=n then return 0;
    else return T(n-1,k) + T(n-2,k-1);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jan 25 2020

function T(n,k)
  if k eq 0 then return 1;
  elif k eq n and n lt 4 then return 1;
  elif k eq n then return 0;
  else return T(n-1,k) + T(n-2, k-1);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 25 2020

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=n and n<4 then 1
    elif k=n then 0
    else T(n-1, k) + T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Jan 25 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n && n<4, 1, If[k==n, 0, T[n-1, k] + T[n-2, k-1] ]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 25 2020 *)

T(n,k) = if(k==0, 1, if(k==n && n<4, 1, if(k==n, 0, T(n-1, k) + T(n-2, k-1) )));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 25 2020

@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==n and n<4): return 1
    elif (k==n): return 0
    else: return T(n-1, k) + T(n-2, k-1)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 25 2020

Typo in definition corrected by Georg Fischer, Dec 03 2021

A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 11, 15, 15, 11, 5, 1, 16, 26, 30, 26, 16, 6, 1, 22, 42, 56, 56, 42, 22, 7, 1, 29, 64, 98, 112, 98, 64, 29, 8, 1, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10, 1, 56, 176, 385, 627, 792, 792, 627

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 05 2006

Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).

Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - Paul Barry, Apr 08 2011

			Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).
First few rows of the triangle are
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4,  1;
  11, 15, 15, 11,  5,  1;
  16, 26, 30, 26, 16,  6,  1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1, 1;
   1, 1, 1;
   0, 0, 1, 1;
  -1, 0, 0, 1, 1;
   0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 1, 1;
  -1, 0, 0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
(End)

Cf. A098574, A000125, A055795, A027660, A055796, A002663.

Cf. A000325.

T:=(n,k)->binomial(n,k)+binomial(n,k+2): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Flatten[Table[Binomial[n,k]+Binomial[n,k+2],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 12 2015 *)

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 12 2014

Edited by N. J. A. Sloane, Nov 29 2006

A362193 Number of Grassmannian permutations of size n that avoid a pattern, sigma, where sigma is a pattern of size 6 with exactly one descent.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 57, 113, 211, 373, 628, 1013, 1574, 2367, 3459, 4929, 6869, 9385, 12598, 16645, 21680, 27875, 35421, 44529, 55431, 68381, 83656, 101557, 122410, 146567, 174407, 206337, 242793, 284241, 331178, 384133, 443668, 510379, 584897

Offset: 0

Jessica A. Tomasko, Apr 10 2023

A permutation is said to be Grassmannian if it has at most one descent. The definition for sigma is a pattern of size 6 with exactly one descent. For example, sigma can be chosen to be 124356, 241356, 361245, 512346, etc.

Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, Enum. Combin. Appl. 2 (2022), no. 4, Article #S4PP6.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A027660, A361270, A361271, A356185, A000325, A027927.

a:= n-> 1+(n-1)*n*(n+1)*(n*(n-5)+26)/120:
seq(a(n), n=0..38);  # Alois P. Heinz, Apr 12 2023

CoefficientList[Series[(1 - 5 x + 11 x^2 - 12 x^3 + 7 x^4 - x^5)/(1 - x)^6, {x, 0, 38}], x] (* Michael De Vlieger, Apr 12 2023 *)

a(n) = 1 + sum(i=3, 6, binomial(n, i-1)) \\ Andrew Howroyd, Apr 10 2023

a(n) = 1 + Sum_{i=2..5} binomial(n,i).
G.f.: (1-5*x+11*x^2-12*x^3+7*x^4-x^5)/(1-x)^6.
a(0) = 1; a(1) = 1; a(n) = 1 + A027660(n-2), n >= 2. - Omar E. Pol, Apr 12 2023

cp's OEIS Frontend

A080851 Square array of pyramidal numbers, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Derive

Maple

Mathematica

Formula

A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

A055794 Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=1 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j) = T(i-1,j) + T(i-2,j-1) for 1<=j<=i-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Extensions

A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A362193 Number of Grassmannian permutations of size n that avoid a pattern, sigma, where sigma is a pattern of size 6 with exactly one descent.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula