A117468 -id:A117468 - cp's OEIS frontend

A117469 The largest part summed over all partitions of n in which every integer from the smallest part to the largest part occurs.

1, 3, 6, 9, 13, 19, 24, 30, 42, 49, 61, 79, 92, 110, 144, 162, 195, 242, 278, 332, 405, 463, 546, 656, 759, 882, 1049, 1205, 1399, 1655, 1887, 2181, 2546, 2909, 3361, 3880, 4422, 5069, 5831, 6641, 7566, 8666, 9818, 11159, 12730, 14376, 16281, 18465, 20828

Offset: 1

Emeric Deutsch, Mar 19 2006

nonn

a(n)=Sum(k*A117468(n,k),k=1..n).

			a(5)=13 because in the 5 (=A034296(5)) partitions in which every integer from the smallest to the largest part occurs, namely [5],[3,2],[2,2,1],[2,1,1,1] and [1,1,1,1,1], the sum of the largest parts is 5+3+2+2+1=13.

Cf. A117466, A117467, A117468.

g:=sum(x^j*product(1+x^i,i=1..j-1)*(1+(1-x^j)*sum(x^i/(1+x^i),i=1..j-1))/(1-x^j)^2,j=1..70): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..55);

G.f.=sum(x^j*product(1+x^i, i=1..j-1)*[1+(1-x^j)sum(x^i/(1+x^i), i=1..j-1)]/(1-x^j)^2, j=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117468 and setting t=1).

A275662 Triangle read by rows: T(n,k) = number of convex domino towers with n dominoes having widest row with k dominoes.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 15, 18, 7, 1, 31, 48, 17, 9, 1, 63, 109, 49, 20, 11, 1, 127, 240, 115, 52, 24, 13, 1, 255, 498, 258, 122, 61, 28, 15, 1, 511, 1026, 551, 261, 136, 71, 32, 17, 1, 1023, 2065, 1163, 531, 298, 157, 81, 36, 19, 1

Offset: 1

Tricia Muldoon Brown, Aug 04 2016

A domino tower is built by placing dominoes horizontally on a convex horizontal base. A domino tower is convex if all its columns and rows are convex.

			Triangle begins:
1;
3, 1;
7, 6, 1;
15, 18, 7, 1;
...
If n = 3 and k = 2, the widest row of the domino tower has two dominoes. Thus the third domino may be found supporting the row of two dominoes in one way or being supported by the row of two dominoes in 5 ways, so T(3,2) = 6.

T. M. Brown, Convex domino towers, arXiv:1608.01562 [math.CO] (2016).

Column 1: A000225, n>=1.

Cf. A117468, A275204, A275599.

G.f.: (2*A_k(x)+B_k(x))*(C_{k-1}(x)+1) where A_k(x) is the generating function on right-skewed domino towers with a base of k dominoes from the sequence A275599, B_k(x) is the generating function on domino stacks with a base of k dominoes associated with the sequence A275204, and C_k(x) is the generating function on flat partitions whose largest part is k-1 given by the sequence A117468.

cp's OEIS Frontend

A117469 The largest part summed over all partitions of n in which every integer from the smallest part to the largest part occurs.

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