A355440 -id:A355440 - cp's OEIS frontend

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

1, 2, 4, 17, 210, 9217, 1399298, 811229225, 2071392232962, 20710319937493889, 1137259214532706572162, 255141201504146525745627265, 348787971214016591166179037803522, 2262996819897931095524655885144485185409

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A086331, A320287, A349893, A355440, A355463.

Cf. A000248, A135746, A355473.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^k*x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*x^k/k!)))

a(n) = sum(k=0, n, k^(k*(n-k))*binomial(n, k));

E.g.f.: Sum_{k>=0} exp(k^k * x) * x^k/k!.

a(n) = Sum_{k=0..n} k^(k*(n-k)) * binomial(n,k).

A360934 Expansion of e.g.f. Sum_{k>=0} exp((4^k - 1)x) x^k/k!.

Original entry on oeis.org

1, 1, 7, 73, 1711, 75121, 6743287, 1169659513, 412296162271, 284887781497441, 400134611520973927, 1108533158650520901673, 6238465090832886119430031, 69421876683500992783472318161, 1567475216919199483376363835235927

Offset: 0

Seiichi Manyama, Feb 26 2023

Cf. A001831, A360933, A360935.

Cf. A135754, A355440.

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(4^k-1)*x)^(k+1)))

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

G.f.: Sum_{k>=0} x^k/(1 - (4^k - 1)*x)^(k+1).

a(n) = Sum_{k=0..n} (4^k - 1)^(n-k) * binomial(n,k).

A355395 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(j(n-j)) binomial(n,j).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 6, 8, 2, 1, 2, 8, 26, 16, 2, 1, 2, 10, 56, 162, 32, 2, 1, 2, 12, 98, 704, 1442, 64, 2, 1, 2, 14, 152, 2050, 15392, 18306, 128, 2, 1, 2, 16, 218, 4752, 84482, 593408, 330626, 256, 2, 1, 2, 18, 296, 9506, 318752, 7221250, 39691136, 8488962, 512, 2

Offset: 0

Seiichi Manyama, Jul 02 2022

The Stanley reference below describes a family of binomial posets whose elements are two colored graphs with vertices labeled on [n] and with edges labeled on [k-1]. T(n,k) is the number of elements in an n-interval of such a binomial poset. - Geoffrey Critzer, Aug 21 2023

			Square array begins:
  1,  1,    1,     1,     1,      1, ...
  2,  2,    2,     2,     2,      2, ...
  2,  4,    6,     8,    10,     12, ...
  2,  8,   26,    56,    98,    152, ...
  2, 16,  162,   704,  2050,   4752, ...
  2, 32, 1442, 15392, 84482, 318752, ...

R. P. Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Example 3.18.3(e), page 323.

Columns k=0..4 give A040000, A000079, A047863, A135079, A355440.
Main diagonal gives A320287.
Cf. A009999.

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

E.g.f. of column k: Sum_{j>=0} exp(k^j * x) * x^j/j!.
G.f. of column k: Sum_{j>=0} x^j/(1 - k^j * x)^(j+1).
For k>=1, E(x)^2 = Sum_{n>=0} T(n,k)*x^n/B_k(n) where B_k(n) = n!*k^binomial(n,2) and E(x) = Sum_{n>=0} x^n/b_k(n). - Geoffrey Critzer, Aug 21 2023

cp's OEIS Frontend

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A360934 Expansion of e.g.f. Sum_{k>=0} exp((4^k - 1)x) x^k/k!.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A355395 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(j(n-j)) binomial(n,j).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A360934 Expansion of e.g.f. Sum_{k>=0} exp((4^k - 1)*x) * x^k/k!.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A355395 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(j*(n-j)) * binomial(n,j).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Formula

A360934 Expansion of e.g.f. Sum_{k>=0} exp((4^k - 1)x) x^k/k!.

A355395 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(j(n-j)) binomial(n,j).