

A265249


Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).


2



1, 2, 3, 5, 7, 10, 1, 13, 2, 17, 4, 1, 20, 8, 2, 26, 11, 4, 1, 29, 17, 8, 2, 35, 24, 13, 4, 1, 39, 33, 19, 8, 2, 48, 39, 30, 13, 4, 1, 48, 56, 41, 21, 8, 2, 60, 64, 57, 32, 13, 4, 1, 61, 83, 75, 47, 21, 8, 2, 74, 94, 100, 65, 34, 13, 4, 1
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OFFSET

1,2


COMMENTS

Number of entries in row n is floor((n4)/2) (n>=4).
Sum of entries of row n = A000041(n) = number of partitions of n.
T(n,0) = A265250(n).
Sum(k*T(n,k), k>=0) = A182977(n).


LINKS

Table of n, a(n) for n=1..67.


FORMULA

G.f.: G(t,x) = Sum_{i>=1} x^i/(1x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/(1x^i)/(1x^j)/Product_{k=i+1..j1} (1tx^k).


EXAMPLE

T(8,2) = 1 because among the 22 partitions of 8 only [3,2,2,1] has 2 parts strictly between the smallest and the largest part.
Triangle starts:
1;
2;
3;
5;
7;
10, 1;
13, 2;


MAPLE

g := add(x^i/(1x^i), i=1..80)+add(add(x^(i+j)/((1x^i)*(1x^j)*mul(1t*x^k, k=i+1..j1)), j=i+1..80), i=1..80): gser := simplify(series(g, x=0, 23)): for n to 22 do P[n]:= sort(coeff(gser, x, n)) end do: for n to 22 do seq(coeff(P[n], t, k), k=0..degree(P[n])) end do; # yields sequence in triangular form


CROSSREFS

CF. A000041, A182977, A265250.
Sequence in context: A280429 A076387 A193622 * A214331 A182483 A308818
Adjacent sequences: A265246 A265247 A265248 * A265250 A265251 A265252


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Dec 25 2015


STATUS

approved



