1,24

a(N,k) with N=A000217(n-1) + m, where A000217(n-1) is the largest triangular number less than N, is the number of partitions of n into m parts which have the parts of the k-th partition of m (in Abramowitz-Stegun order) as exponents.

The sequence of row lengths of this array is p(m)= A000041(m) (number of partitions of m) and m is determined from N (the row index) as explained above. It is [1,1,2,1,2,3,1,2,3,5,1,2,3,5,7,1,2,3,5,7,11,...]=A092080(N), N>=1.

One can find the (n,m; k) numbers for the p-th entry (p>2) of the sequence as follows: p= a(n-1) + b(m-1) + k, where a(n-1) := A085360(n-1) is the largest number from the numbers A085360 less than p, and b(m-1)=A026905(m-1) is the largest number from the numbers A026905 less than p-a(n-1). p=1 belongs to (1,1;1) and p=2 to (2,1;1).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, pp. 831-2.

W. Lang, First 36 rows and more comments.

N=13 = 10 + 3 with 10=A000217(4), hence n=5 and m=3.

N=10 = 6 + 4 with 6=A000217(3), hence n=4 and m=4.

The sequence entry nr. p=16, which is 0, belongs to (n=4,m=3; k=3)

because 16 = 10 + 3 + 3 with 10=A085360(3), hence n=4, and 3=A026905(2),

hence m=3.

a(N=13,k=2)=2, n=5, m=3; there are exactly 2 partitions of 5 into 3 parts, each having the parts of the second (k=2) partition of 3, i.e. 1,2, as exponents. These two 3-partitions of 5 are: [1^2, 3^1] and [1^1, 2^2], which are all the 3-partitions of 5 because the other entries of row N=13 are 0.

Cf. A092079.

Sequence in context: A062590 A139215 A139216 this_sequence A067109 A030219 A035147

Adjacent sequences: A092075 A092076 A092077 this_sequence A092079 A092080 A092081

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 19 2004