A305466 -id:A305466 - cp's OEIS frontend

A305401 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..floor(n/2)} ((n-j)!/j!)binomial(n-j,j)k^(n-2*j).

1, 1, 0, 1, 1, 1, 1, 2, 3, 0, 1, 3, 9, 10, 1, 1, 4, 19, 56, 43, 0, 1, 5, 33, 174, 457, 225, 1, 1, 6, 51, 400, 2107, 4626, 1393, 0, 1, 7, 73, 770, 6433, 31779, 55969, 9976, 1, 1, 8, 99, 1320, 15451, 129060, 574129, 788192, 81201, 0, 1, 9, 129, 2086, 31753, 387045, 3103873, 12088488, 12667041, 740785, 1

Offset: 0

Seiichi Manyama, Jun 02 2018

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
   0,  1,   2,    3,    4,     5, ...
   1,  3,   9,   19,   33,    51, ...
   0, 10,  56,  174,  400,   770, ...
   1, 43, 457, 2107, 6433, 15451, ...

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

Columns k=0-3 give A059841, A001040(n+1), A036243, A305459.
Rows n=0-2 give A000012, A001477, A058331.
Main diagonal gives A305465.
Cf. A305466.

A(n,k) = k*n*A(n-1,k) + A(n-2,k) for n>1.

A305467 a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)binomial(n-k,k)n^(n-2k)(-1)^k.

Original entry on oeis.org

1, 1, 7, 150, 5857, 363045, 32817311, 4078256168, 667231014401, 139047475691385, 35961972186044999, 11303290914120251574, 4243674498966718214113, 1875719852330658989518045, 964140893268009386931042943, 570249392860305817156465883040

Offset: 0

Seiichi Manyama, Jun 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Main diagonal of A305466.

Cf. A305465.

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

a(n) ~ n! * n^n. - Vaclav Kotesovec, Jun 03 2018

A305471 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) - a(n-2).

Original entry on oeis.org

1, 3, 17, 150, 1783, 26595, 476927, 9988872, 239256001, 6449923155, 193258438649, 6371078552262, 229165569442783, 8931086129716275, 374876451878640767, 16860509248409118240, 808929567471759034753, 41238547431811301654163

Offset: 0

Seiichi Manyama, Jun 02 2018

nonn

Let S(i,j,n) denote a sequence of the form a(0) = 1, a(1) = i, a(n) = i*n*a(n-1) + j*a(n-2). Then S(i,j,n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*i^(n-2*k)*j^k.

Seiichi Manyama, Table of n, a(n) for n = 0..380

Column k=3 of A305466.

Cf. A305459, A305472.

{a(n) = sum(k=0, n/2, ((n-k)!/k!)*binomial(n-k, k)*3^(n-2*k)*(-1)^k)}

a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*3^(n-2*k)*(-1)^k.

a(n) ~ BesselJ(0, 2/3) * n! * 3^n. - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

A305401 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..floor(n/2)} ((n-j)!/j!)binomial(n-j,j)k^(n-2*j).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A305467 a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)binomial(n-k,k)n^(n-2k)(-1)^k.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A305471 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) - a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A305401 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..floor(n/2)} ((n-j)!/j!)*binomial(n-j,j)*k^(n-2*j).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A305467 a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*n^(n-2*k)*(-1)^k.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A305471 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) - a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A305401 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..floor(n/2)} ((n-j)!/j!)binomial(n-j,j)k^(n-2*j).

A305467 a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)binomial(n-k,k)n^(n-2k)(-1)^k.

A305471 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) - a(n-2).