A294253 -id:A294253 - cp's OEIS frontend

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 13, 1, 0, 1, 1, 3, 19, 49, 1, 0, 1, 1, 3, 19, 97, 261, 1, 0, 1, 1, 3, 19, 121, 681, 1531, 1, 0, 1, 1, 3, 19, 121, 921, 5971, 9073, 1, 0, 1, 1, 3, 19, 121, 1041, 8491, 50443, 63393, 1, 0, 1, 1, 3, 19, 121, 1041

Offset: 0

Seiichi Manyama, Oct 26 2017

			Square array A(n,k) begins:
   1, 1,   1,   1,   1, ...
   0, 1,   1,   1,   1, ...
   0, 1,   3,   3,   3, ...
   0, 1,  13,  19,  19, ...
   0, 1,  49,  97, 121, ...
   0, 1, 261, 681, 921, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000012, A118589, A294251, A294252, A294253.
Rows n=0 gives A000012.
Main diagonal gives A293840.
Cf. A070936, A294212, A294254.

B(j,k) is the coefficient of Product_{i=1..k} (1+x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(A000217(k),n)} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).

Original entry on oeis.org

1, 1, 3, 19, 121, 1041, 11191, 130663, 1731969, 25778161, 432791371, 7752723771, 151553121193, 3178030999729, 71244609480591, 1716351868658911, 43661944977384961, 1173984102030774753, 33302371396771085779, 991402105480284394531, 30912472614894951462681

Offset: 0

Alois P. Heinz, Sep 21 2019

nonn

Differs from A293840 and from A294253 first at n = 6.

			a(3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A327673.

Cf. A293840, A294253.

b:= proc(n, i, k, p) option remember;
     `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
    end:
a:= n-> add(b(n$2, i, 0)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..21);

b[n_, i_, k_, p_] := b[n, i, k, p] =
     If[n == 0, p!, If[i < 1, 0, Sum[Binomial[k^i, j]*
     b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, i, 0] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

a(n) = A327673(n,n).

cp's OEIS Frontend

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

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A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).

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