A307862 -id:A307862 - cp's OEIS frontend

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1+4k)*x^2).

1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -3, -5, 1, 1, 1, -5, -11, -5, 1, 1, 1, -7, -17, 1, 11, 1, 1, 1, -9, -23, 19, 81, 41, 1, 1, 1, -11, -29, 49, 211, 141, 29, 1, 1, 1, -13, -35, 91, 401, 181, -363, -125, 1, 1, 1, -15, -41, 145, 651, 41, -2015, -1791, -365, 1

Offset: 0

Seiichi Manyama, May 02 2019

			Square array begins:
   1,  1,    1,     1,     1,      1,      1, ...
   1,  1,    1,     1,     1,      1,      1, ...
   1, -1,   -3,    -5,    -7,     -9,    -11, ...
   1, -5,  -11,   -17,   -23,    -29,    -35, ...
   1, -5,    1,    19,    49,     91,    145, ...
   1, 11,   81,   211,   401,    651,    961, ...
   1, 41,  141,   181,    41,   -399,  -1259, ...
   1, 29, -363, -2015, -5767, -12459, -22931, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Columns k=0..5 give A000012, A098331, A098332, A098333, A098334.
Main diagonal gives A307862.
Cf. A307819, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + x - k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).

A307911 Coefficient of x^n in expansion of (1 - nx - nx^2)^n.

Original entry on oeis.org

1, -1, 0, 27, -416, 5625, -74304, 924385, -8626176, -48361131, 7124800000, -340421390199, 13686496542720, -522760216822129, 19658830846298112, -735037915447265625, 27218267709730979840, -980444996625142158435, 32830565919734078521344, -889052809376495994642527

Offset: 0

Seiichi Manyama, May 05 2019

sign

Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*n*x + n*(n+4)*x^2).

Seiichi Manyama, Table of n, a(n) for n = 0..386

Main diagonal of A307819.

Cf. A092366, A307862.

a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * Binomial[2*k, k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Join[{1}, Table[(-n)^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -4/n], {n, 1, 20}]] (* Vaclav Kotesovec, May 12 2021 *)

PARI
```
{a(n) = polcoef((1-n*x-n*x^2)^n, n)}
```

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

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

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

For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 1, -4/n). - Vaclav Kotesovec, May 12 2021

A307947 Coefficient of x^n in 1/(n+1) * (1 + x - n*x^2)^(n+1).

Original entry on oeis.org

1, 1, -1, -8, 9, 201, -89, -8721, -5599, 540595, 1091551, -43580206, -159753527, 4318835223, 24185472039, -506659112324, -3987724324735, 68460979242171, 722884820238847, -10431188525128096, -143862807220127799, 1761112732849258195, 31284729872945906919

Offset: 0

Seiichi Manyama, May 07 2019

sign

Also coefficient of x^n in the expansion of 2/(1 - x + sqrt(1 - 2*x + (1+4*n)*x^2)).

Robert Israel, Table of n, a(n) for n = 0..594

Cf. A000108, A307862, A307906.

f:= n -> coeff(1/(n+1)*(1+x-n*x^2)^(n+1),x,n):
map(f, [$0..30]); # Robert Israel, May 08 2019

a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 23, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, -4*n], {n, 0, 20}] (* Vaclav Kotesovec, May 12 2021 *)

{a(n) = polcoef((1+x-n*x^2)^(n+1)/(n+1), n)}

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

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

a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,2*k) * A000108(k).

a(n) = Hypergeometric2F1(1/2 - n/2, -n/2, 2, -4*n). - Vaclav Kotesovec, May 12 2021

cp's OEIS Frontend

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1+4k)*x^2).

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A307911 Coefficient of x^n in expansion of (1 - nx - nx^2)^n.

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A307947 Coefficient of x^n in 1/(n+1) * (1 + x - n*x^2)^(n+1).

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cp's OEIS Frontend

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).

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A307911 Coefficient of x^n in expansion of (1 - n*x - n*x^2)^n.

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A307947 Coefficient of x^n in 1/(n+1) * (1 + x - n*x^2)^(n+1).

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Author

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Maple

Mathematica

PARI

PARI

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Formula

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1+4k)*x^2).

A307911 Coefficient of x^n in expansion of (1 - nx - nx^2)^n.