A136406 -id:A136406 - cp's OEIS frontend

A137504 Limiting values of A136406: a(n) = A136406(m,m-n) for any m >= 2n.

1, 1, 3, 4, 8, 11, 21, 28, 47, 64, 100, 135, 205, 273, 398, 530, 749, 989, 1373, 1796, 2446, 3183, 4264, 5509, 7294, 9357, 12245, 15623, 20234, 25663, 32964, 41569, 52970, 66472, 84090, 105006, 132013, 164072, 205052, 253770, 315426, 388749, 480846, 590276, 726741

Offset: 0

Benoit Jubin, Apr 22 2008

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..250

Cf. A136406.

P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
seq(n)={my(m=2*n); Vec(polcoef(prod(w=1, sqrtint(m), sum(k=0, m\w^2, (x^w*y)^(k*w) / P(k,w^2,m))), m))[1..n]} \\ Andrew Howroyd, Oct 23 2019

Terms a(21) and beyond from Andrew Howroyd, Oct 22 2019

A140188 Table read by rows: T(n,k) is the number of groupoids (categories all of whose morphisms are invertible) with n morphisms and k objects.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 1, 1, 2, 4, 5, 3, 1, 1, 1, 5, 6, 5, 3, 1, 1, 5, 8, 10, 9, 5, 3, 1, 1, 2, 10, 14, 12, 9, 5, 3, 1, 1, 2, 13, 21, 20, 15, 9, 5, 3, 1, 1, 1, 13, 24, 29, 23, 15, 9, 5, 3, 1, 1, 5, 20, 39, 42, 37, 27, 15, 9, 5, 3, 1, 1, 1, 19, 43, 58, 53, 40, 27, 15, 9, 5, 3, 1, 1, 2

Offset: 1

Benoit Jubin, May 12 2008

The first column is T(n,1) = A000001(n) (number of groups of order n).
T(n,k) >= A136406(n,k).
The sum of the n^th row is A140189(n).
For 2k<=n, T(n,n-k) = A140190(k) does not depend on n.

Cf. A140185.

T(n,k) is the sum over the quadratic bi-partitions (n_i,k_i) of (n,k) (see A136406) of the "product" of the A000001(n_i), where the "product" is the usual product except when (n_i1,k_i1)=...=(n_ip,k_ip), in which case a^p is replaced by binomial(a+p-1,p).

A136405 Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i = n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 4, 3, 7, 1, 4, 6, 7, 5, 11, 1, 3, 7, 8, 11, 7, 15, 1, 5, 8, 16, 14, 17, 11, 22, 1, 4, 12, 14, 23, 21, 25, 15, 30, 1, 6, 12, 24, 29, 38, 33, 37, 22, 42, 1, 5, 15, 24, 41, 42, 57, 47, 52, 30, 56, 1, 7, 18, 37, 47, 75, 68, 87, 70, 74, 42, 77

Offset: 1

Benoit Jubin, Apr 13 2008

			Triangle begins:
  1;
  1, 2;
  1, 1,  3;
  1, 3,  2,  5;
  1, 2,  4,  3,  7;
  1, 4,  6,  7,  5, 11;
  1, 3,  7,  8, 11,  7, 15;
  1, 5,  8, 16, 14, 17, 11, 22;
  1, 4, 12, 14, 23, 21, 25, 15, 30;
  1, 6, 12, 24, 29, 38, 33, 37, 22, 42;
  ...
T(4,2) = 3 since (4,2) can be bi-partitioned as (2,2) or ((1,1),(3,1)) or ((2,1),(2,1)).
T(5,3) = 4 since (5,3) can be bi-partitioned as ((1,1),(2,2)) or ((3,1),(1,2)) or ((1,1),(1,1),(3,1)) or ((1,1),(2,1),(2,1)).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A006171.

Cf. A000041, A028242, A090806, A136406.

P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, n, sum(k=0, n\w, (x*y)^(k*w)/P(k,w,n))), n)/y)}
{ for(n=1, 10, print(T(n))) } \\ Andrew Howroyd, Oct 23 2019

T(n,1) = 1.
T(n,2) = A028242(n).
T(n,n) = A000041(n).

Terms a(57) and beyond from Andrew Howroyd, Oct 23 2019

cp's OEIS Frontend

A137504 Limiting values of A136406: a(n) = A136406(m,m-n) for any m >= 2n.

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A140188 Table read by rows: T(n,k) is the number of groupoids (categories all of whose morphisms are invertible) with n morphisms and k objects.

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A136405 Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i = n.

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