A258462 Number of partitions of n into parts of exactly 7 sorts which are introduced in ascending order.

1, 29, 492, 6401, 70880, 704676, 6490951, 56524414, 471750267, 3810085912, 29989229859, 231255237311, 1754111872429, 13128442913712, 97189645384884, 713050007285941, 5192646586465458, 37581376345088462, 270593146237918806, 1939929376872664097

Offset: 7

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 7..1000

Crossrefs

Column k=7 of A256130.

Cf. A320549.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,7):
seq(a(n), n=7..30);

Mathematica

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[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
Table[T[n, 7], {n, 7, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 7^n, where c = 1/(7!*Product_{n>=1} (1-1/7^n)) = 1/(7!*QPochhammer[1/7, 1/7]) = 0.0002371101666331046535758625585353... . - Vaclav Kotesovec, Jun 01 2015