A289546 -id:A289546 - cp's OEIS frontend

A289946 a(n) = Sum_{k=1..n} k!^6.

1, 65, 46721, 191149697, 2986175149697, 139317055679149697, 16390300280131775149697, 4296598745804900241599149697, 2283384320190476620685217983149697, 2283382306976051006261597069217983149697

Offset: 1

Eric W. Weisstein, Jul 16 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..104
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018. [This thesis cites this sequence entry, but it's just a typo: the intended sequence entry is A289546.]
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A007489 (k!), A104344 (k!^2), A289945 (k!^4).

Cf. A289947 (indices giving primes).

Table[Sum[k!^6, {k, n}], {n, 10}]
Accumulate[(Range[10]!)^6] (* Harvey P. Dale, May 14 2023 *)

a(n) = sum(k=1, n, k!^6); \\ Michel Marcus, Jul 16 2017

A293845 Triangle read by rows: T(n,k) is the number of chains of length k in the partially ordered (by subspace inclusion) set of all subspaces of GF(2)^n, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 5, 7, 3, 16, 50, 56, 21, 67, 446, 1010, 945, 315, 374, 5395, 22692, 40455, 32550, 9765, 2825, 92881, 704601, 2167179, 3193155, 2255715, 615195, 29212, 2350136, 32061404, 162602418, 394534644, 496062000, 312519060, 78129765, 417199, 89342600, 2220570872, 18194735010, 68980503390, 138302085600, 151794972000

Offset: 0

Geoffrey Critzer, Oct 17 2017

			Triangle begins:
1;
2, 1;
5, 7, 3;
16, 50, 56, 21;
67, 446, 1010, 945, 315;
374, 5395, 22692, 40455, 32550, 9765;
...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Cf. A289546, A293844 (row sums), A005329 (main diagonal), A006116 (column k = 0).

nn = 10; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,
  Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,
     nn}] CoefficientList[Series[ eq[z]^2/(1 - u (eq[z] - 1)) /. q -> 2, {z, 0, nn}], {z, u}]]]

T(n,k)/A005329(n) is the coefficient of y^k*x^n in eq(x)^2/(1 - y (eq(x) - 1)) where eq(x) is the q-exponential function.

cp's OEIS Frontend

A289946 a(n) = Sum_{k=1..n} k!^6.

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