A305459 -id:A305459 - 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.

A305460 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) + 2*a(n-2).

Original entry on oeis.org

1, 3, 20, 186, 2272, 34452, 624680, 13187184, 317741776, 8605402320, 258797553152, 8557530058656, 308588677217920, 12052073471616192, 506804263162315904, 22830295989247448064, 1096867816010202138880, 55985919208498803979008

Offset: 0

Seiichi Manyama, Jun 01 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. A222467, A305459.

List([0..20],n->Sum([0..Int(n/2)],k->((Factorial(n-k))/(Factorial(k))*Binomial(n-k,k)*3^(n-2*k)*2^k))); # Muniru A Asiru, Jun 01 2018

a:=proc(n) option remember: if n=0 then 1 elif n=1 then 3 elif n>=2 then 3*n*procname(n-1)+2*procname(n-2) fi; end:
seq(a(n),n=0..20); # Muniru A Asiru, Jun 01 2018

nxt[{n_,a_,b_}]:={n+1,b,3b(n+1)+2a}; NestList[nxt,{1,1,3},20][[;;,2]] (* Harvey P. Dale, Feb 08 2025 *)

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

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

a(n) ~ BesselI(0, 2*sqrt(2)/3) * n! * 3^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).

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A305460 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) + 2*a(n-2).

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

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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

A305460 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) + 2*a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

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).

A305460 a(0) = 1, a(1) = 3, a(n) = 3na(n-1) + 2*a(n-2).

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