cp's OEIS Frontend

A047913 Triangle of numbers a(n,k) = number of partitions of k such that k = n + n_1 + n_2 + ... + n_t where n_1 <= 2n and n_{i+1} <= 2n_i for all i.

%I A047913 #23 Jan 15 2024 12:02:32
%S A047913 1,1,1,1,1,2,1,1,2,3,1,1,2,4,5,1,1,2,4,7,9,1,1,2,4,7,12,16,1,1,2,4,7,
%T A047913 13,22,28,1,1,2,4,7,13,24,39,50,1,1,2,4,7,13,24,42,70,89,1,1,2,4,7,13,
%U A047913 24,43,76,126,159,1,1,2,4,7,13,24,43,78,137,225,285
%N A047913 Triangle of numbers a(n,k) = number of partitions of k such that k = n + n_1 + n_2 + ... + n_t where n_1 <= 2n and n_{i+1} <= 2n_i for all i.
%C A047913 Triangle is read in this order: a(1,1), a(2,2), a(1,2), a(3,3), a(2,3), a(1,3), a(4,4), ...
%C A047913 Rows are the columns in the table at the end of the Minc reference, read bottom to top. - _Joerg Arndt_, Jan 15 2024
%H A047913 H. Minc, <a href="http://dx.doi.org/10.1017/S0013091500021945">A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid</a>, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
%F A047913 a(n, n)=1, a(n, k) = Sum_{i=1..2n} a(i, k-n).
%e A047913 Triangle begins:
%e A047913 1;
%e A047913 1, 1;
%e A047913 1, 1, 2;
%e A047913 1, 1, 2, 3;
%e A047913 1, 1, 2, 4, 5;
%e A047913 1, 1, 2, 4, 7,  9;
%e A047913 1, 1, 2, 4, 7, 12, 16;
%e A047913 1, 1, 2, 4, 7, 13, 22, 28;
%e A047913 1, 1, 2, 4, 7, 13, 24, 39, 50;
%e A047913 1, 1, 2, 4, 7, 13, 24, 42, 70,  89;
%e A047913 1, 1, 2, 4, 7, 13, 24, 43, 76, 126, 159;
%e A047913 1, 1, 2, 4, 7, 13, 24, 43, 78, 137, 225, 285;
%e A047913 ...
%e A047913 Rows approach A002843. - _Joerg Arndt_, Jan 15 2024
%t A047913 a[n_, n_] = 1; a[n_?Positive, k_?Positive] := a[n, k] = Sum[a[i, k-n], {i, 1, 2*n}]; a[n_, k_] = 0; Table[a[n, k], {k, 1, 12}, {n, k, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 21 2013 *)
%Y A047913 Rows give A002572, A002573, A002574, ..., columns approach A002843.
%Y A047913 Cf. A049286 (triangle with reversed rows).
%K A047913 tabl,nonn,easy,nice
%O A047913 1,6
%A A047913 _N. J. A. Sloane_