A055796 -id:A055796 - cp's OEIS frontend

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

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.

A080852 Square array of 4D pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 15, 20, 1, 7, 20, 35, 35, 1, 8, 25, 50, 70, 56, 1, 9, 30, 65, 105, 126, 84, 1, 10, 35, 80, 140, 196, 210, 120, 1, 11, 40, 95, 175, 266, 336, 330, 165, 1, 12, 45, 110, 210, 336, 462, 540, 495, 220, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 286

Offset: 0

Paul Barry, Feb 21 2003

The first row contains the tetrahedral numbers, which are really three-dimensional, but can be regarded as degenerate 4D pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

			Array, n >= 0, k >= 0, begins
1 4 10 20  35  56 ...
1 5 15 35  70 126 ...
1 6 20 50 105 196 ...
1 7 25 65 140 266 ...
1 8 30 80 175 336 ...

Index to sequences related to pyramidal numbers

Rows include A000292, A000332, A002415, A001296, A002418, A002419, A051740, A051797.
Cf. A057145, A080851, A180266, A055796 (antidiagonal sums).
See A257200 for another version of the array.

vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^5,x,11),x,n),n,0,11),k,-1,10)

VECTOR(VECTOR(comb(k+3,3)+comb(k+3,4)n, k, 0, 11), n, 0, 11)

A080852 := proc(n,k)
        binomial(k+4,4)+(n-1)*binomial(k+3,4) ;
end proc:
seq( seq(A080852(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

T[n_, k_] := Binomial[k+3, 3] + Binomial[k+3, 4]n;
Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

T(n, k) = binomial(k + 4, 4) + (n-1)*binomial(k + 3, 4), corrected Oct 01 2021.
T(n, k) = T(n - 1, k) + C(k + 3, 4) = T(n - 1, k) + k(k + 1)(k + 2)(k + 3)/24.
G.f. for rows: (1 + nx)/(1 - x)^5, n >= -1.
T(n,k) = sum_{j=0..k} A080851(n,j). - R. J. Mathar, Jul 28 2016

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

A125230 Triangle T(n,k) (0<=k<=n) read by rows in which column k contains the binomial transform of the sequence of k 0's, (k+1) 1's, followed by 0's.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 11, 5, 1, 1, 15, 25, 16, 6, 1, 1, 21, 50, 42, 22, 7, 1, 1, 28, 91, 98, 64, 29, 8, 1, 1, 36, 154, 210, 163, 93, 37, 9, 1, 1, 45, 246, 420, 381, 256, 130, 46, 10, 1, 1, 55, 375, 792, 837, 638, 386, 176, 56, 11, 1, 1, 66, 550, 1419, 1749, 1485, 1024

Offset: 0

Gary W. Adamson, Nov 24 2006

A125231 is another triangle with the same row sums A045623: (1, 2, 5, 12, 28, 64, 144, 320...).

			T(5,2) = C(5,2) + C(5,3) + C(5,4) = 10 + 10 + 5 = 25.
First few rows of the triangle are:
1
1 1
1 3 1
1 6 4 1
1 10 11 5 1
1 15 25 16 6 1

Cf. A007318, A125231. Columns k=0-3 give: A000012, A000217, A006522(n+1), A055796(n-3). Row sums give: A045623.

T:= (n, k)-> add (binomial (n, j), j=k..min(2*k, n)): seq (seq (T(n, k), k=0..n), n=0..12);

T(n,k) = Sum_{j=k..min(2*k,n)} C(n,j).

Edited with more terms and Maple program by Alois P. Heinz, Oct 16 2009

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

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A080852 Square array of 4D pyramidal numbers, read by antidiagonals.

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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.

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

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A125230 Triangle T(n,k) (0<=k<=n) read by rows in which column k contains the binomial transform of the sequence of k 0's, (k+1) 1's, followed by 0's.

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