A253830 -id:A253830 - cp's OEIS frontend

A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

1, 1, 2, 4, 7, 12, 20, 33, 53, 84, 131, 202, 308, 465, 695, 1030, 1514, 2209, 3201, 4609, 6596, 9386, 13284, 18705, 26211, 36561, 50776, 70226, 96742, 132765, 181540, 247369, 335940, 454756, 613689, 825698, 1107755, 1482038, 1977465, 2631664

Offset: 0

Paul D. Hanna, Dec 31 2006

nonn

In triangle A126347, row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Row sums of A253830. a(n) equals the number of colored compositions of n, as defined in A253830, whose associated color partition has distinct parts. An example is given below. - Peter Bala, Jan 20 2015

			a(5) = 12: The colored compositions (defined in A253830) of 5 whose color partitions have distinct parts are
5(c1), 5(c2), 5(c3), 5(c4), 5(c5),
1(c1) + 4(c2), 1(c1) + 4(c3), 1(c1) + 4(c4),
3(c1) + 2(c2),
2(c1) + 3(c2), 2(c1) + 3(c3), 2(c2) + 3(c3). - _Peter Bala_, Jan 20 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from Seiichi Manyama)

Cf. A126347, A126349; factorial variant: A126471. A253830, A307599, A307601, A307602.

nmax = 50; CoefficientList[Series[Product[(1 - x + x^k)/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2019 *)

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
{a(n)=Vec(B(n+1,'q)+O('q^(n*(n-1)/2+1)))[n*(n-1)/2+1]}

{a(n) = local(t); if( n<0, 0, t = 1; polcoeff( sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = t * x^k / (1 - x) / (1 - x^k) + x * O(x^n), 1), n))} /* Michael Somos, Aug 17 2008 */

1 + Sum_{k>0} x^(k * (k + 1) / 2) / ((1 - x)^k * (1 - x) * (1 - x^2) ... (1 - x^k)). - Michael Somos, Aug 17 2008
G.f.: Product_{k>0} (1+x^k/(1-x)). - Vladeta Jovovic, Oct 05 2008
G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)/(d*(1 - x)^d)). - Ilya Gutkovskiy, Apr 19 2019

A253829 Triangular array with g.f. Product_{n >= 1} 1/(1 - x*z^n/(1 - z)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 7, 4, 1, 0, 5, 13, 11, 5, 1, 0, 6, 22, 25, 16, 6, 1, 0, 7, 34, 50, 41, 22, 7, 1, 0, 8, 50, 91, 92, 63, 29, 8, 1, 0, 9, 70, 155, 187, 155, 92, 37, 9, 1, 0, 10, 95, 250, 353, 343, 247, 129, 46, 10, 1, 0, 11, 125, 386, 628, 701, 590, 376, 175, 56, 11, 1

Offset: 0

Peter Bala, Jan 19 2015

A refinement of A227682.
A colored composition of n is defined as a composition of n where each part p comes in one of p colors (denoted by an integer from 1 to p) and the color numbers are nondecreasing through the composition. The color numbers thus form a partition, called the color partition, of some integer.
For example, the composition 1 + 3 + 2 of 6 gives rise to three colored compositions of 6, namely, 1(c1) + 3(c1) + 2(c1), 1(c1) + 3(c1) + 2(c2) and 1(c1) + 3(c2) + 2(c2), where the color number of a part is shown after the part prefaced by the letter c.
T(n,k) equals the number of colored compositions of n into k parts.
See A253830 for the enumeration of colored compositions having parts with distinct colors.

			Triangle begins
n\k| 0  1   2   3   4   5   6  7
= = = = = = = = = = = = = = = = =
0  | 1
1  | 0  1
2  | 0  2   1
3  | 0  3   3   1
4  | 0  4   7   4   1
5  | 0  5  13  11   5   1
6  | 0  6  22  25  16   6  1
7  | 0  7  34  50  41  22  7  1
...
T(4,2) = 7: The compositions of 4 into two parts are 2 + 2, 1 + 3 and 3 + 1. Coloring the parts as described above produces seven colored compositions of 4 into two parts:
2(c1) + 2(c1), 2(c1) + 2(c2), 2(c2) + 2(c2),
1(c1) + 3(c1), 1(c1) + 3(c2), 1(c1) + 3(c3),
3(c1) + 1(c1).

P. Bala, Colored Compositions

Cf. A008284, A227682 (row sums), A253830.

G := 1/(product(1-x*z^j/(1-z), j = 1 .. 12)): Gser := simplify(series(G, z = 0, 14)): for n to 12 do P[n] := coeff(Gser, z^n) end do: for n to 12 do seq(coeff(P[n], x^j), j = 1 .. n) end do;

G.f.: G(x,z) := Product_{n >= 1} (1 - z)/(1 - z - x*z^n) = exp( Sum_{n >= 1} (x*z)^n/(n*(1 - z)^n*(1 - z^n)) ) =
1 + Sum_{n >= 1} (x*z/(1 - z))^n/( Product_{i = 1..n} 1 - z^i ) = 1 + x*z + (2*x + x^2)*z^2 + (3*x + 3*x^2 + x^3)*z^3 + ....
Note, G(x*(1 - z),z) is the generating function of A008284.
T(n,k) = Sum_{i = k..n} binomial(i-1,k-1)*A008284(n+k-i,k).
Recurrence equation: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-k,k) - T(n-k-1,k) with boundary conditions T(n,n) = 1, T(n,0) = 0 for n >= 1 and T(n,k) = 0 for n < k.
Row sums are A227682.

cp's OEIS Frontend

A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

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