A361280 -id:A361280 - cp's OEIS frontend

A361283 Expansion of e.g.f. exp(x/(1-x)^4).

1, 1, 9, 85, 961, 13041, 207001, 3746149, 75832065, 1693615681, 41302616041, 1090835399061, 30988423000129, 941461990360945, 30439632977968761, 1042973073239321701, 37731609890300935681, 1436586994020158747649

Offset: 0

Seiichi Manyama, Mar 06 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414

Column k=4 of A293012.

Cf. A361280.

A361283 := proc(n)
    n!*add(binomial(n+3*k-1,n-k)/k!,k=0..n) ;
end proc:
seq(A361283(n),n=0..40) ; # R. J. Mathar, Mar 12 2023

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^4)))

a(n) = n!*sum(k=0, n, (-1)^(n-k)*binomial(-4*k, n-k)/k!);

a(n) = n!*sum(k=0, n, binomial(n+3*k-1, n-k)/k!);

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, (-1)^(j-1)*j*binomial(-4, j-1)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(-4*k,n-k)/k! = n! * Sum_{k=0..n} binomial(n+3*k-1,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} (-1)^(k-1) * k * binomial(-4,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) +(-5*n+4)*a(n-1) +(n-1)*(10*n-23)*a(n-2) -10*(n-1)*(n-2)*(n-3)*a(n-3) +5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) ~ 2^(1/5) * n^(n - 1/10) * exp(-27/1280 - 13*2^(3/5)*n^(1/5)/800 + 13*2^(1/5)*n^(2/5)/240 + 2^(-6/5)*n^(3/5) + 5*2^(-8/5)*n^(4/5) - n) / sqrt(5) * (1 + 116303*2^(12/5)/(3200000*n^(1/5))). - Vaclav Kotesovec, Nov 11 2023

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 19, 25, 1, 1, 1, 9, 37, 97, 81, 1, 1, 1, 11, 61, 241, 581, 331, 1, 1, 1, 13, 91, 481, 1981, 3661, 1303, 1, 1, 1, 15, 127, 841, 4881, 17551, 26335, 5937, 1, 1, 1, 17, 169, 1345, 10001, 55321, 171697, 202049, 26785, 1

Offset: 0

Seiichi Manyama, Mar 06 2023

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  1,   1,    1,    1,     1, ...
  1,  3,   5,    7,    9,    11, ...
  1,  7,  19,   37,   61,    91, ...
  1, 25,  97,  241,  481,   841, ...
  1, 81, 581, 1981, 4881, 10001, ...

Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.
Main diagonal gives A361281.
Cf. A293012.

T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);

E.g.f. of column k: exp(x * (1+x)^k).

T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.

A361569 Expansion of e.g.f. exp(x^4/24 * (1+x)^4).

Original entry on oeis.org

1, 0, 0, 0, 1, 20, 180, 840, 1715, 2520, 88200, 1940400, 29111775, 303603300, 2188286100, 12549537000, 143029511625, 3397035642000, 71419225878000, 1170096883956000, 15075357741068625, 163540869094102500, 2025016641129982500, 40912918773391665000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A047974, A361567, A361568.

Cf. A361280.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*(1+x)^4)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, j*binomial(4, j-4)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(4*k,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * binomial(4,k-4) * a(n-k)/(n-k)!.
a(n) = (n-1)*(n-2)*(n-3)/24 * (4*a(n-4) + 20*(n-4)*a(n-5) + 36*(n-4)*(n-5)*a(n-6) + 28*(n-4)*(n-5)*(n-6)*a(n-7) + 8*(n-4)*(n-5)*(n-6)*(n-7)*a(n-8)). -Seiichi Manyama, Jun 16 2024

cp's OEIS Frontend

A361283 Expansion of e.g.f. exp(x/(1-x)^4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

PARI

PARI

PARI

Formula

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A361569 Expansion of e.g.f. exp(x^4/24 * (1+x)^4).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula