A292462 - OEIS

%I #25 May 19 2018 12:36:13

%S 1,1,5,31,278,3287,48256,843567,17081639,392869430,10112244792,

%T 287927207846,8984122319997,304828239096197,11173376516829974,

%U 439988449921648076,18523908107054523591,830292183207722271065,39475390430795389762048,1984220622132901208082220

%N Number of partitions of n with n sorts of part 1.

%H Alois P. Heinz, <a href="/A292462/b292462.txt">Table of n, a(n) for n = 0..386</a>

%F a(n) = [x^n] 1/(1-n*x) * Product_{j=2..n} 1/(1-x^j).

%F a(n) ~ n^n * (1 + 1/n^2 + 1/n^3 + 2/n^4 + 2/n^5 + 4/n^6 + 4/n^7 + 7/n^8 + 8/n^9 + 12/n^10), for coefficients see A002865. - _Vaclav Kotesovec_, Sep 19 2017

%F a(n) = Sum_{j=0..n} A002865(j) * n^(n-j). - _Alois P. Heinz_, Sep 22 2017

%e a(2) = 5: 2, 1a1a, 1a1b, 1b1a, 1b1b.

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

%p       `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))

%p     end:

%p a:= n-> b(n$3):

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

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^n, If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];

%t a[0] = 1; a[n_] := b[n, n, n];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 19 2018, translated from Maple *)

%Y Cf. A002865, A246935, A292463, A292503, A292507, A292567.

%Y Main diagonal of A292741.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 16 2017