A124305 -id:A124305 - cp's OEIS frontend

A124304 Riordan array (1, x*(1-x^2)).

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

Offset: 0

Paul Barry, Oct 25 2006

T(2n,n) is a signed aerated version of C(2n,n).

Inverse is A124305.

			Triangle begins
  1;
  0,  1;
  0,  0,  1;
  0, -1,  0,  1;
  0,  0, -2,  0,  1;
  0,  0,  0, -3,  0,  1;
  0,  0,  1,  0, -4,  0,  1;
  0,  0,  0,  3,  0, -5,  0,  1;
  0,  0,  0,  0,  6,  0, -6,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

Cf. A014021 (diagonal sums), A050935 (row sums), A124305 (inverse).

Cf. A002054, A079977, A126869, A138364, A176971.

A124304:= func< n,k | (&+[(-1)^j*Binomial(k,k-j)*Binomial(k,n-k-j) : j in [0..n]]) >;
[A124304(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;
Table[A124304[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2
flatten([[A124304(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.
T(n, k) = C(k,(n-k)/2)*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A050935(n+2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).
T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).
From G. C. Greubel, Aug 18 2023: (Start)
T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).
T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).
G.f.: 1/(1 - x*y*(1-x^2)). (End)

A124790 A generalized Motzkin triangle.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 3, 4, 3, 2, 1, 0, 6, 9, 6, 5, 2, 1, 0, 15, 21, 15, 12, 6, 3, 1, 0, 36, 51, 36, 30, 15, 9, 3, 1, 0, 91, 127, 91, 76, 40, 25, 10, 4, 1, 0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1

Offset: 0

Paul Barry, Nov 07 2006

Columns include A005043, A001006, A002026. Row sums are A124791. For even k, column k has g.f. x^k*M(x)^(k/2), where M(x)=2/(1-x+sqrt(1-2x-3x^2)) is the g.f. of A001006. For odd k, column k has g.f. x^k*S(x)*M(x)^floor(k/2), S(x)=(1+x-sqrt(1-2x-3x^2))/(2x(1+x)), the g.f. of A005043.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 1, 2, 1, 1,
0, 3, 4, 3, 2, 1,
0, 6, 9, 6, 5, 2, 1,
0, 15, 21, 15, 12, 6, 3, 1,
0, 36, 51, 36, 30, 15, 9, 3, 1,
0, 91, 127, 91, 76, 40, 25, 10, 4, 1,
0, 232, 323, 232, 196, 105, 69, 29, 14, 4, 1
Production matrix begins
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 1, 1, 1, 1, 1, 1, 1, 1
- _Paul Barry_, Apr 07 2011

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Triangle is the product of A124788 and A124305, that is, it is the product of the expansion of (1+x*y)/(1-x^2*y^2-x^3*y^2) and the inverse of the Riordan array (1,x(1-x^2)).

A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 3, 3, 3, 2, 1, 0, 0, 7, 4, 5, 2, 1, 0, 12, 12, 12, 10, 6, 3, 1, 0, 0, 30, 18, 24, 12, 9, 3, 1, 0, 55, 55, 55, 50, 32, 22, 10, 4, 1, 0, 0, 143, 88, 121, 66, 57, 25, 14, 4, 1, 0, 273, 273, 273

Offset: 0

Paul Barry, Nov 08 2006

Product of A124305 and A124788. Columns include aerated A001764,A047749(n+1),A124817,A084081. Row sums are A124818.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 2, 1, 1,
0, 3, 3, 3, 2, 1,
0, 0, 7, 4, 5, 2, 1,
0, 12, 12, 12, 10, 6, 3, 1,
0, 0, 30, 18, 24, 12, 9, 3, 1,
0, 55, 55, 55, 50, 32, 22, 10, 4, 1

Cf. A124790.

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A124304 Riordan array (1, x*(1-x^2)).

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A124790 A generalized Motzkin triangle.

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A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

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