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Number of compositions of n^2 into parts >= n.
5

%I #23 Mar 04 2022 11:01:49

%S 1,1,2,6,26,140,882,6349,51284,457704,4459940,47019819,532485538,

%T 6438774524,82710138994,1123798871990,16090426592488,241979954659728,

%U 3811335657375786,62712512310820402,1075527196672980525,19186234784992217621,355349469934379290700

%N Number of compositions of n^2 into parts >= n.

%H Alois P. Heinz, <a href="/A332796/b332796.txt">Table of n, a(n) for n = 0..170</a>

%e a(0) = 1: (), the empty composition.

%e a(1) = 1: 1.

%e a(2) = 2: 22, 4.

%e a(3) = 6: 333, 36, 63, 45, 54, 9.

%e a(4) = 26: 4444, 556, 565, 655, 466, 646, 664, 457, 475, 547, 574, 745, 754, 448, 484, 844, 88, 79, 97, 6(10), (10)6, 5(11), (11)5, 4(12), (12)4, (16).

%p b:= proc(n, k) option remember; `if`(n=0, 1,

%p add(b(n-j, k), j=k..n))

%p end:

%p a:= n-> b(n^2, n):

%p seq(a(n), n=0..23);

%t b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k], {j, k, n}]];

%t a[n_] := b[n^2, n];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Mar 04 2022, after _Alois P. Heinz_ *)

%Y Cf. A011782, A103488, A332716, A332721, A332747.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 24 2020