A026729 -id:A026729 - cp's OEIS frontend

A125250 Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 5, 1, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 3, 11, 3, 0, 0, 0, 0, 0, 0, 1, 13, 13, 1, 0, 0, 0, 0, 0, 0, 0, 9, 26, 9, 0, 0, 0, 0, 0, 0, 0, 0, 4, 32, 32, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, 26, 63, 26, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 80, 80, 14, 0, 0, 0, 0, 0

Offset: 1

Gerald McGarvey, Jan 15 2007

It appears that the main diagonal (1,1,2,5,11,...) is A051286 (Whitney number of level n of the lattice of the ideals of the fence of size 2 n) that the diagonals (0,1,2,5,13,...) adjacent to the main diagonal are A110320 (Number of blocks in all RNA secondary structures with n nodes) and that the n-th antidiagonal sum = A094686(n-1) (a Fibonacci convolution). The n-th row sum = A002605(n).

			Array starts as:
1 0 0 0  0  0  0 ...
0 1 1 0  0  0  0 ...
0 1 2 2  1  0  0 ...
0 0 2 5  5  3  1   0 ...
0 0 1 5 11 13  9   4   1   0...
0 0 0 3 13 26 32  26  14   5   1  0 ...
0 0 0 1  9 32 63  80  71  45  20  6  1 0 ...
0 0 0 0  4 26 80 153 201 191 135 71 27 7 1 0 ...
...

Cf. A051286, A110320, A002605, A026729, A030528, A057089, A109466.

T[n_, k_] := Sum[Binomial[i, n-i] Binomial[i, k-i], {i, Floor[(n+1)/2], k}];
Table[T[n-k, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 12 2019 *)

A=matrix(22,22);A[1,1]=1;A[2,2]=1;A[2,1]=0;A[1,2]=0;A[3,2]=1;A[2,3]=1; for(n=3,22,for(k=3,22,A[n,k]=A[n-2,k-2]+A[n-1,k-2]+A[n-2,k-1]+A[n-1,k-1])); for(n=1,22,for(i=1,n,print1(A[n-i+1,i],", ")))

A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).
From Peter Bala, Nov 07 2017: (Start)
T(n,k) = Sum_{i = floor((n+1)/2)..k} binomial(i,n-i)* binomial(i,k-i).
Square array = A026729 * transpose(A026729), where A026729 is viewed as a lower unit triangular array. Omitting the first row and column of square array = A030528 * transpose(A030528).
O.g.f. 1/(1 - t*(1 + t)*x - t*(1 + t)*x^2) = 1 + (t + t^2)*x + (t + 2*t^2 + 2*t^3 + t^4)*x^2 + .... Cf. A109466 with o.g.f. 1/(1 - t*x - t*x^2).
The n-th row polynomial R(n,t) satisfies R(n,t) = R(n,-1 - t).
R(n,t) = (-1)^n*sqrt(-t*(1 + t))^n*U(n, 1/2*sqrt(-t*(1 + t))), where U(n,x) denotes the n-th Chebyshev polynomial of the second kind.
The sequence of row polynomials R(n,t) is a divisibility sequence of polynomials, that is, if m divides n then R(m,t) divides R(n,t) in the polynomial ring Z[t].
R(n,1) = A002605; R(n,2) = A057089. (End)

A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 1, 0, 0, -1, 1, 4, 3, 0, 0, 0, 1, 1, 5, 6, 1, 0, 0, 0, -1, 1, 6, 10, 4, 0, 0, 0, 0, 1, 1, 7, 15, 10, 1, 0, 0, 0, 0, -1, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, -1, 1, 10, 36

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

Differs from A164925 in signs.
The n-th row consists of the coefficients in the expansion of (-x)^n + (((1 + sqrt(1 + 4*x))/2)^n -((1 - sqrt(1 + 4*x))/2)^n )/sqrt(1 + 4*x).
The coefficients in the expansion of Sum_{j=0..floor((n - 1)/2)} T(n,k)*x^(n - 2*j - 1) yield the n-th row in A168561, the coefficients of the n-th Fibonacci polynomial.
Row n sums up to Fibonacci(n) + (-1)^n (A008346).

			Triangle begins:
    1;
    1, -1;
    1,  0,  1;
    1,  1,  0, -1;
    1,  2,  0,  0,  1;
    1,  3,  1,  0,  0, -1;
    1,  4,  3,  0,  0,  0, 1;
    1,  5,  6,  1,  0,  0, 0, -1;
    1,  6, 10,  4,  0,  0, 0,  0, 1;
    1,  7, 15, 10,  1,  0, 0,  0, 0, -1;
    1,  8, 21, 20,  5,  0, 0,  0, 0,  0, 1;
    1,  9, 28, 35, 15,  1, 0,  0, 0,  0, 0, -1;
    ...

Wikipedia, Fibonacci polynomials

Inspired by A123018.

Cf. A007318, A026729, A049310, A052553, A164925, A168561.

Table[Table[Binomial[n - k - 1, k], {k, 0, n}], {n, 0, 12}]//Flatten

create_list(binomial(n - k - 1, k), n, 0, 12, k, 0, n);

G.f.: 1/((1 + x*y)*(1 - y - x*y^2)).
E.g.f.: exp(-x*y) + (exp(y*(1 + sqrt(1 + 4*x))/2) - exp(y*(1 - sqrt(1 + 4*x))/2))/sqrt(1 + 4*x).
T(n,1) = A023443(n).

A099092 Riordan array (1,2+4x).

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 0, 0, 16, 8, 0, 0, 16, 48, 16, 0, 0, 0, 96, 128, 32, 0, 0, 0, 64, 384, 320, 64, 0, 0, 0, 0, 512, 1280, 768, 128, 0, 0, 0, 0, 256, 2560, 3840, 1792, 256, 0, 0, 0, 0, 0, 2560, 10240, 10752, 4096, 512, 0, 0, 0, 0, 0, 1024, 15360, 35840, 28672, 9216, 1024, 0, 0, 0

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A063727. Diagonal sums are A052907.

The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

			Rows begin
  {1},
  {0,  2},
  {0,  4,  4},
  {0,  0, 16,  8},
  {0,  0, 16, 48, 16}, ...

Cf. A026729, A038210, A052907, A063727, A113953.

Number triangle T(n,k) = binomial(k, n-k)*2^n; columns have g.f. (2x+4x^2)^k.

T(n,k) = A113953(n,k)*2^k = A026729(n,k)*2^n. - Philippe Deléham, Dec 11 2008

A104731 Triangle T(n,k) = sum_{j=k..n} (j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 5, 11, 4, 1, 5, 16, 19, 5, 1, 5, 16, 37, 29, 6, 1, 5, 16, 44, 71, 41, 7, 1, 5, 16, 44, 103, 121, 55, 8, 1, 5, 16, 44, 112, 211, 190, 71, 9, 1, 5, 16, 44, 112, 261, 390, 281, 89, 10, 1, 5, 16, 44, 112, 272, 555, 666, 397, 109, 11

Offset: 0

Gary W. Adamson, Mar 20 2005

			The first few rows of the triangle are:
1;
1, 2;
1, 5, 3;
1, 5, 11, 4
1, 5, 16, 19, 5;
1, 5, 16, 37, 29, 6;
...

Cf. A014286 (row sums), A045925, A026729.

Product of the triangles A(n,k) = k+1 and B = binomial(k,n-k) = [1; 0, 1; 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...], the triangular view of A026729.

A177040 Irregular triangle t(n,m) = binomial(m+1,n-m) read by rows floor((n+1)/2) <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 4, 1, 6, 5, 1, 4, 10, 6, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, 21, 70, 84, 45, 11, 1, 7, 56, 126, 120, 55, 12, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 36, 210, 462, 495, 286, 91, 15, 1

Offset: 0

Roger L. Bagula, May 01 2010

Row sums are in A052952.

Contains the right half of each row of A030528. - R. J. Mathar, May 19 2013

			1;
1;
2, 1;
3, 1;
3, 4, 1;
6, 5, 1;
4, 10, 6, 1;
10, 15, 7, 1;
5, 20, 21, 8, 1;
15, 35, 28, 9, 1;
6, 35, 56, 36, 10, 1;
21, 70, 84, 45, 11, 1;
7, 56, 126, 120, 55, 12, 1;
28, 126, 210, 165, 66, 13, 1;
8, 84, 252, 330, 220, 78, 14, 1;
36, 210, 462, 495, 286, 91, 15, 1;

Cf. A180987 (read diagonally downwards), A098925, A026729, A085478, A165253

t[n_, m_] := Binomial[m + 1, n - m];
Table[Table[t[n, m], {m, Floor[(n + 1)/2], n}], {n, 0, 15}];
Flatten[%]

T(m,n)=binomial(n+1,m-n) \\ Charles R Greathouse IV, May 19 2013

cp's OEIS Frontend

A125250 Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

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A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

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A099092 Riordan array (1,2+4x).

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A104731 Triangle T(n,k) = sum_{j=k..n} (j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

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A177040 Irregular triangle t(n,m) = binomial(m+1,n-m) read by rows floor((n+1)/2) <= m <= n.

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