A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

1, 1, 2, 129, 2316, 94510, 4939220, 211106945, 14879165560, 828070125876, 61472962084968, 4223017425122958, 325536754765395096, 25399546083773839692, 2059386837863675003112, 173281152533121109073025, 14789443838781868027714800, 1307994690673355979749969800

Offset

0,3

Comments

For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(7,n)}. Gould (1974) conjectured that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).

a(n) is the total number of 7-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2). - Alois P. Heinz, Apr 02 2023

Links

Table of n, a(n) for n=0..17.

H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Formula

a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^7.

From Alois P. Heinz, Apr 02 2023: (Start)

a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^7.

a(n) = Sum_{j=0..n} A120730(n,j)^7.

a(n) = A357824(n,7). (End)

a(n) ~ 3 * 2^(7*n + 27/2) / (2401 * Pi^(7/2) * n^(13/2)). - Vaclav Kotesovec, Aug 27 2023

Maple

seq(add( ( binomial(n, k) - binomial(n, k-1) )^7, k = 0..floor(n/2)), n = 0..20);

Mathematica

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^7, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2023 *)

Crossrefs

Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A183069 ( S(3,2*n+1)/ S(1,2*n+1) ), A361887 ( S(5,n) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/ S(1,2*n-1) ).

Column k=7 of A357824.

Cf. A008315, A120730.

Sequence in context: A342618 A258806 A216358 * A294274 A099824 A351916

Adjacent sequences: A361887 A361888 A361889 * A361891 A361892 A361893

Keyword

nonn,easy

Author

Peter Bala, Mar 30 2023

Status

approved