cp's OEIS Frontend

A092078 Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents.

%I A092078 #18 Apr 01 2025 03:38:37
%S A092078 1,1,1,0,1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0,1,0,2,0,2,0,0,1,0,0,0,1,0,
%T A092078 0,0,0,0,0,1,1,2,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
%U A092078 1,0,3,0,3,1,0,2,0,1,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0
%N A092078 Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents.
%C A092078 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.
%C A092078 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.
%C A092078 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).
%H A092078 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
%H A092078 Wolfdieter Lang, <a href="/A092078/a092078.txt">First 36 rows and more comments</a>.
%e A092078 N=13 = 10 + 3 with 10=A000217(4), hence n=5 and m=3.
%e A092078 N=10 = 6 + 4 with 6=A000217(3), hence n=4 and m=4.
%e A092078 The sequence entry nr. p=16, which is 0, belongs to (n=4,m=3; k=3)
%e A092078 because 16 = 10 + 3 + 3 with 10=A085360(3), hence n=4 and 3=A026905(2),
%e A092078 hence m=3.
%e A092078 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.
%Y A092078 Cf. A092079.
%K A092078 nonn,easy,tabf
%O A092078 1,24
%A A092078 _Wolfdieter Lang_, Mar 19 2004