A361600 -id:A361600 - cp's OEIS frontend

A361599 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x).

1, 2, 11, 88, 881, 10526, 145867, 2294636, 40302593, 780263866, 16483592171, 376901809472, 9265228770481, 243493769839958, 6808261249400171, 201697053847178836, 6308214318127014017, 207622266953125336946, 7170928402389293540683, 259247888385780787392296

Offset: 0

Seiichi Manyama, Mar 17 2023

nonn

Column k=3 of A361600.

Cf. A091695.

Table[n! * Sum[Binomial[n+2*k,3*k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)))

a(n) = n! * Sum_{k=0..n} binomial(n+2*k,3*k)/k! = Sum_{k=0..n} (n+2*k)!/(3*k)! * binomial(n,k).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = 2*(2*n - 1)*a(n-1) - (n-1)*(6*n - 11)*a(n-2) + (n-2)*(n-1)*(4*n - 9)*a(n-3) - (n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(-1/8) * exp(-1/27 - 3^(-5/4)*n^(1/4)/8 + sqrt(n/3)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 1/8) / 2 * (1 + (34237/69120)*3^(1/4)/n^(1/4)). (End)

A361598 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

Original entry on oeis.org

1, 2, 9, 58, 473, 4626, 52537, 677594, 9762993, 155175778, 2693718281, 50657791482, 1025158123849, 22198908725618, 511885585833273, 12517101011344666, 323402336324055137, 8800318580852865474, 251497162228635927433, 7529081846683064675258

Offset: 0

Seiichi Manyama, Mar 17 2023

nonn

Column k=2 of A361600.

Cf. A082579.

Table[n! * Sum[Binomial[n+k,2*k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^2)/(1-x)))

a(n) = n! * Sum_{k=0..n} binomial(n+k,2*k)/k! = Sum_{k=0..n} (n+k)!/(2*k)! * binomial(n,k).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 1)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/6) * (1 + 1/(2^(2/3)*n^(1/3)) + 83/(360*2^(1/3)*n^(2/3))). (End)

A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 15, 34, 1, 1, 5, 25, 103, 209, 1, 1, 6, 37, 214, 885, 1546, 1, 1, 7, 51, 373, 2293, 9051, 13327, 1, 1, 8, 67, 586, 4721, 29176, 106843, 130922, 1, 1, 9, 85, 859, 8481, 70981, 427189, 1425495, 1441729, 1

Offset: 0

Seiichi Manyama, Mar 18 2023

			Square array begins:
  1,    1,    1,     1,     1,      1, ...
  1,    2,    3,     4,     5,      6, ...
  1,    7,   15,    25,    37,     51, ...
  1,   34,  103,   214,   373,    586, ...
  1,  209,  885,  2293,  4721,   8481, ...
  1, 1546 ,9051, 29176, 70981, 146046, ...

Columns k=0..3 give A000012, A002720, A343884, A351767.
Main diagonal gives A361617.
Cf. A293985, A361600.

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

E.g.f. of column k: exp( x/(1-x)^k ) / (1-x)^k.

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

A361607 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)k,nk)/k!.

Original entry on oeis.org

1, 2, 9, 88, 1457, 35226, 1158097, 49554464, 2664907233, 175012404562, 13725980234201, 1263867766626312, 134795551989905809, 16464112185873351338, 2280346417134518709537, 355060682992984062716176

Offset: 0

Seiichi Manyama, Mar 17 2023

nonn

Main diagonal of A361600.

Cf. A293013.

Table[Sum[Binomial[n, k]*(n*k + n - k)!/(n*k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)

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

a(n) = n! * [x^n] exp( x/(1-x)^n ) / (1-x).
a(n) = Sum_{k=0..n} (n*k+n-k)!/(n*k)! * binomial(n,k).
log(a(n)) ~ n*(2*log(n) - log(log(n)) - 1 - log(2) + (log(log(n)) + log(2) + 1/2)/log(n)). - Vaclav Kotesovec, Mar 17 2023

cp's OEIS Frontend

A361599 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x).

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A361598 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

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A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

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Formula

A361607 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)k,nk)/k!.

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cp's OEIS Frontend

A361599 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A361598 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A361616 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

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PARI

Formula

A361607 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)*k,n*k)/k!.

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Formula

A361607 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)k,nk)/k!.