A117569 -id:A117569 - cp's OEIS frontend

A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 19, 10, 1, 1, 15, 45, 45, 15, 1, 1, 21, 90, 141, 90, 21, 1, 1, 28, 161, 357, 357, 161, 28, 1, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55

Offset: 1

Colin Mallows, Aug 23 2000

Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - Paul D. Hanna, Feb 26 2005

Subtriangle (for 1<=k<=n) of triangle defined by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2006

			Triangle begins:
  1;
  1,1;
  1,3,1;
  1,6,6,1;
  1,10,19,10,1;
  ...
Triangle (0, 1, 0, 1, 0, 0, 0...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3, 1;
  0, 1, 6, 6, 1;
  0, 1, 10, 19, 10, 1;
  0, 1, 15, 45, 45, 15, 1;
  0, 1, 21, 90, 141, 90, 21, 1;
  ... - _Philippe Deléham_, Mar 27 2014

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 6.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 7.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

Columns are A000217, A005712, A005714, A005716.

Cf. A027907, A104027, A117569, A124302.

t[n_, k_] := Sum[ Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Oct 11 2011, after Paul Barry *)

PARI
```
T(n,k)=if(nPaul D. Hanna
```

T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).
G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - Paul D. Hanna, Feb 26 2005
Sum_{k, 1<=k<=n}T(n,k)=A124302(n). Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n). - Philippe Deléham, Oct 29 2006
From Paul Barry, Sep 28 2010: (Start)
G.f.: 1/(1-x-xy-x^2y/(1-x-xy)).
E.g.f.: exp((1+y)x)*cosh(sqrt(y)*x).
T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j)). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,1) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<1 or if k>n. - Philippe Deléham, Mar 27 2014

More terms from James Sellers, Aug 25 2000

More terms from Paul D. Hanna, Feb 26 2005

A163805 Expansion of (1 - x) * (1 - x^6) / ((1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0

Offset: 0

Michael Somos, Aug 04 2009

			G.f. = 1 - x + x^3 - x^5 + x^7 - x^9 + x^11 - x^13 + x^15 - x^17 + x^19 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A117569, A163806.

1, seq(sin(3*n*Pi/2), n=1..100); # Ridouane Oudra, Nov 18 2024

a[ n_] := Boole[n == 0] + {-1, 0, 1, 0}[[Mod[n, 4, 1]]]; (* Michael Somos, Sep 06 2015 *)

{a(n) = (n==0) + [0, -1, 0, 1][n%4 + 1]};

PARI
```
{a(n) = (n==0) - kronecker(-4, n)};
```

Euler transform of length 6 sequence [ -1, 0, 1, 1, 0, -1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v) - u * (2 - u) * (3 - 2*v).
a(2*n) = 0 unless n=0, a(4*n + 1) = -1, a(4*n + 3) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+4) = a(n) unless n=0 or n=-4.
a(n) = - A117569(n) unless n=0. a(n) = (-1)^n * A117569(n).
Convolution inverse of A163806.
G.f.: (1 - x + x^2) / (1 + x^2).
G.f. A(x) = 1 - x / (1 + x^2) = 1 / (1 + x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Jan 03 2013
a(n) = A101455(n-2) = A056594(n-3), n>2. - R. J. Mathar, Aug 06 2009
E.g.f.: 1 - sin(x). - Stefano Spezia, Nov 16 2024
a(n) = sin(3*n*Pi/2), for n>0. - Ridouane Oudra, Nov 18 2024

A117568 Riordan array ((1-x^3)/(1+x^2),x).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, 1, -1, -1, 0, 1, -1, 1, 1, -1, -1, 0, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, 1, -1, -1, 1, 1, -1, -1, 0, 1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 0, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 0, 1

Offset: 0

Paul Barry, Mar 29 2006

Inverse of number triangle A117567. Row sums are A117569. Diagonal sums are A102560.

			Triangle begins
1,
0, 1,
-1, 0, 1,
-1, -1, 0, 1,
1, -1, -1, 0, 1,
1, 1, -1, -1, 0, 1,
-1, 1, 1, -1, -1, 0, 1,
-1, -1, 1, 1, -1, -1, 0, 1,
1, -1, -1, 1, 1, -1, -1, 0, 1

A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 19, 10, 1, 0, 1, 15, 45, 45, 15, 1, 0, 1, 21, 90, 141, 90, 21, 1, 0, 1, 28, 161, 357, 357, 161, 28, 1, 0, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 0, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 0, 1, 55, 615, 2850, 6765, 8953

Offset: 0

Philippe Deléham, Oct 30 2006

Subtriangle (1 <= k <= n) is in A056241.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  6,   6,    1;
  0, 1, 10,  19,   10,    1;
  0, 1, 15,  45,   45,   15,    1;
  0, 1, 21,  90,  141,   90,   21,    1;
  0, 1, 28, 161,  357,  357,  161,   28,    1;
  0, 1, 36, 266,  784, 1107,  784,  255,   36,   1;
  0, 1, 45, 414, 1554, 2907, 2907, 1554,  414,  45,  1;
  0, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55, 1;

Columns: A000007, A000012, A000217, A005712, A005714, A005716.

Cf. A027907, A056241.

T(2*k-1,k) = A082758(k-1)for k >= 1.
Sum_{k=0..n} T(n,k) = A124302(n); see also A007051.
Sum_{k=0..n} (-1)^(n-k)*T(n,k) = A117569(n).
G.f.: (1-x*(y+2)+x^2)/(1-2x*(1+y)+(1+y+y^2)*x^2). - Philippe Deléham, Oct 30 2011

A344566 T(n, k) = (-1)^(n - k)binomial(n - 1, k - 1)hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 1, 1, -3, 1, 0, -1, 2, 3, -4, 1, 0, 0, -4, 2, 6, -5, 1, 0, 1, 2, -9, 0, 10, -6, 1, 0, -1, 3, 9, -15, -5, 15, -7, 1, 0, 0, -6, 3, 24, -20, -14, 21, -8, 1, 0, 1, 3, -18, -6, 49, -21, -28, 28, -9, 1

Offset: 0

Peter Luschny, May 23 2021

The inverse of the Riordan array for directed animals A122896. Without the first column (1, 0, 0, ...) the inverse of the Motzkin triangle A064189.

			Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0, -1,  1;
[3] 0,  0, -2,  1;
[4] 0,  1,  1, -3,   1;
[5] 0, -1,  2,  3,  -4,   1;
[6] 0,  0, -4,  2,   6,  -5,   1;
[7] 0,  1,  2, -9,   0,  10,  -6, 1;
[8] 0, -1,  3,  9, -15,  -5,  15, -7,  1;
[9] 0,  0, -6,  3,  24, -20, -14, 21, -8, 1.

A117569 (row sums).

Cf. A122896, A064189.

T := (n,k) -> (-1)^(n-k)*binomial(n-1,k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], 4): seq(seq(simplify(T(n, k)), k=0..n), n = 0..10);

# uses[riordan_array from A256893]
riordan_array(1, x / (1 + x + x^2), 10)

Riordan_array (1, x / (1 + x + x^2)).

cp's OEIS Frontend

A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

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A163805 Expansion of (1 - x) * (1 - x^6) / ((1 - x^3) * (1 - x^4)) in powers of x.

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A117568 Riordan array ((1-x^3)/(1+x^2),x).

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Examples

A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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Formula

A344566 T(n, k) = (-1)^(n - k)*binomial(n - 1, k - 1)*hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.

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Formula

A344566 T(n, k) = (-1)^(n - k)binomial(n - 1, k - 1)hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.