A305471 -id:A305471 - cp's OEIS frontend

A305466 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-2j)(-1)^j.

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 2, 1, 1, 4, 17, 40, 7, 0, 1, 5, 31, 150, 313, 33, -1, 1, 6, 49, 368, 1783, 3090, 191, 0, 1, 7, 71, 730, 5857, 26595, 36767, 1304, 1, 1, 8, 97, 1272, 14551, 116772, 476927, 511648, 10241, 0, 1, 9, 127, 2030, 30457, 363045, 2796671, 9988872, 8149601, 90865, -1

Offset: 0

Seiichi Manyama, Jun 02 2018

			Square array begins:
    1,  1,   1,    1,    1,     1, ...
    0,  1,   2,    3,    4,     5, ...
   -1,  1,   7,   17,   31,    49, ...
    0,  2,  40,  150,  368,   730, ...
    1,  7, 313, 1783, 5857, 14551, ...

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

Columns k=0-3 give A056594, A058797, A093985(n-1), A305471.
Rows n=0-2 give A000012, A001477, A056220.
Main diagonal gives A305467.
Cf. A305401.

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

A305472 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) - 2*a(n-2).

Original entry on oeis.org

1, 3, 16, 138, 1624, 24084, 430264, 8987376, 214836496, 5782610640, 173048646208, 5699040103584, 204819346436608, 7976556430820544, 334605731401589632, 15041304800209892352, 721313418947271653632, 36756901756710434550528

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

Cf. A305460, A305471.

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

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

cp's OEIS Frontend

A305466 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-2j)(-1)^j.

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A305472 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) - 2*a(n-2).

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cp's OEIS Frontend

A305466 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)^j.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A305472 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) - 2*a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A305466 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-2j)(-1)^j.

A305472 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) - 2*a(n-2).