A258465 - OEIS

%I #16 Dec 07 2020 14:00:07

%S 1,56,1762,41143,795657,13499449,208050040,2979881876,40300054520,

%T 520576172762,6478447651345,78185947269684,919805200917658,

%U 10591351937396242,119764715367192468,1333512940732309728,14652754322423701707,159182411488944508232

%N Number of partitions of n into parts of exactly 10 sorts which are introduced in ascending order.

%C In general, column k>1 of A256130 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - _Vaclav Kotesovec_, Jun 01 2015

%H Alois P. Heinz, <a href="/A258465/b258465.txt">Table of n, a(n) for n = 10..1000</a>

%F a(n) ~ c * 10^n, where c = 1/(10!*Product_{n>=1} (1-1/10^n)) = 1/(10!*QPochhammer[1/10, 1/10]) = 0.0000003096292864992979803727261336621564... . - _Vaclav Kotesovec_, Jun 01 2015

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

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

%p     end:

%p T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):

%p a:= n-> T(n,10):

%p seq(a(n), n=10..30);

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

%t T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];

%t Table[T[n, 10], {n, 10, 30}] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)

%Y Column k=10 of A256130.

%Y Cf. A320552.

%K nonn

%O 10,2

%A _Alois P. Heinz_, May 30 2015