A321878 -id:A321878 - cp's OEIS frontend

A327291 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size eight are used and the colors are introduced in increasing order.

1, 2, 5, 10, 20, 36, 65, 110, 185, 326, 532, 879, 1417, 2272, 3563, 5572, 8543, 13031, 19596, 29671, 43971, 65293, 95783, 140259, 203281, 294069, 421433, 602382, 854470, 1207812, 1700895, 2382536, 3323738, 4619166, 6394401, 8817059, 12117260, 16588535, 22637178

Offset: 36

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 36..5000

Column k=8 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(8):
seq(a(n), n=36..75);

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-7))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-7)) / (4*8!*sqrt(24)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

A327292 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size nine are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 515, 819, 1332, 2102, 3327, 5142, 7958, 12071, 18271, 27256, 40462, 60036, 87981, 128502, 186484, 269466, 386757, 553271, 786299, 1113510, 1568109, 2199730, 3069546, 4278447, 5924730, 8188867, 11266659, 15464516, 21134748

Offset: 45

Alois P. Heinz, Aug 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 45..5000

Column k=9 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(9):
seq(a(n), n=45..83);

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-8))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-8)) / (4*9!*sqrt(27)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

A327293 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size ten are used and the colors are introduced in increasing order.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 795, 1250, 1982, 3087, 4798, 7332, 11191, 16821, 25196, 37308, 54951, 80131, 117346, 169306, 244417, 349967, 500258, 709715, 1005550, 1414751, 1986544, 2773496, 3861747, 5349095, 7389698, 10178856, 13964050, 19102030

Offset: 55

Alois P. Heinz, Aug 28 2019

nonn

In general, for k>=1, is column k of A321878 asymptotic to exp(sqrt(2*(Pi^2 - 6*polylog(2, 1-k))*n/3)) * sqrt(Pi^2 - 6*polylog(2, 1-k)) / (4*k!*sqrt(3*k)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

Alois P. Heinz, Table of n, a(n) for n = 55..5000

Column k=10 of A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(10):
seq(a(n), n=55..93);

a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-9))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-9)) / (4*10!*sqrt(30)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

cp's OEIS Frontend

A327291 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size eight are used and the colors are introduced in increasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A327292 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size nine are used and the colors are introduced in increasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A327293 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size ten are used and the colors are introduced in increasing order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula