A307090 -id:A307090 - cp's OEIS frontend

A307091 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

1, 1, 0, -8, -34, -74, 0, 736, 3334, 7606, 0, -80464, -372436, -864772, 0, 9400192, 43976774, 103061158, 0, -1137528688, -5355697084, -12623082284, 0, 140697113792, 665238165916, 1574005263676, 0, -17663830073504, -83769667651816, -198760191043784, 0

Offset: 0

Seiichi Manyama, Mar 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1879

Central coefficients of number triangle A307090.

Cf. A000984, A119358, A119363.

Table[Sum[(-1)^k*Binomial[n, 2*k]^2, {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 24 2019 *)
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, -1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 24 2019 *)

{a(n) = sum(k=0, n\2, (-1)^k*binomial(n, 2*k)^2)}

a(4*n+2) = 0 for n >= 0.
From Peter Bala, Mar 17 2023: (Start)
n*(n-1)*(6*n^2-24*n+23)a(n) = 4*(n-1)*(2*n-3)*(3*n^2-9*n+4)*a(n-1) - 4*(3*n^2-9*n+4)*(2*n-3)^2*a(n-2) - 8*(n-2)*(2*n-3)*(3*n^2-9*n+4)*a(n-3) - 4*(n-2)*(n-3)*(6*n^2-12*n+5)*a(n-4) with a(0) = 1, a(1) = 1, a(2) = 0 and a(3) = -8.
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], -1).
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 3. (End)

A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3j) binomial(n-k,3*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -3, -3, 1, 1, 1, 1, 1, 1, -9, -15, -9, 1, 1, 1, 1, 1, 1, -19, -39, -39, -19, 1, 1, 1, 1, 1, 1, -34, -79, -99, -79, -34, 1, 1, 1, 1, 1, 1, -55, -139, -199, -199, -139, -55, 1, 1, 1

Offset: 0

Seiichi Manyama, Mar 27 2019

			Triangle begins:
n\k | 0  1  2   3    4   5  6  7  8
----+-------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1, 1;
3   | 1, 1, 1,  1;
4   | 1, 1, 1,  1,   1;
5   | 1, 1, 1,  1,   1,  1;
6   | 1, 1, 1,  0,   1,  1, 1;
7   | 1, 1, 1, -3,  -3,  1, 1, 1;
8   | 1, 1, 1, -9, -15, -9, 1, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

Row sums give A307089.
T(2*n,n) gives A307158.
Cf. A119326, A119335, A307090.

T[n_, k_] := Sum[(-1)^j * Binomial[k, 3*j] * Binomial[n - k, 3*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

cp's OEIS Frontend

A307091 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

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A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3j) binomial(n-k,3*j).

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cp's OEIS Frontend

A307091 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

Views

Author

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Programs

Mathematica

PARI

Formula

A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3*j) * binomial(n-k,3*j).

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Programs

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A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3j) binomial(n-k,3*j).