A103439 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.

0, 1, 3, 7, 16, 39, 105, 315, 1048, 3829, 15207, 65071, 297840, 1449755, 7468541, 40555747, 231335960, 1381989881, 8623700811, 56078446615, 379233142800, 2662013133295, 19362917622001, 145719550012299, 1133023004941272, 9090156910550109, 75161929739797519

Offset: 0

Author

Ralf Stephan, Feb 11 2005

Keywords

• nonn

Comments

Partial sums of A026898.

Antidiagonal sums of array A103438.

Row sums of A123490. - Paul Barry, Oct 01 2006

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..600
• Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.

Crossrefs

Cf. A026898, A103438, A123490.

Programs

Magma

[0] cat [(&+[ (&+[ (k-j+1)^j : j in [0..k]]) : k in [0..n-1]]): n in [1..30]]; // G. C. Greubel, Jun 15 2021

Maple

b:= proc(i) option remember; add((i-j+1)^j, j=0..i) end:
a:= proc(n) option remember; add(b(i), i=0..n-1) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 02 2019

Mathematica

Join[{0},Table[Sum[Sum[(i-j+1)^j,{j,0,i}],{i,0,n}],{n,0,30}]] (* Harvey P. Dale, Dec 03 2018 *)

PARI

a(n) = sum(i=0, n-1, sum(j=0, i, (i-j+1)^j)); \\ Michel Marcus, Jun 15 2021

Sage

[sum(sum((k-j+1)^j for j in (0..k)) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

Formula

a(n+1) = Sum_{k=0..n} ((k+2)^(n-k) + k)/(k+1). - Paul Barry, Oct 01 2006

G.f.: (G(0)-1)/(1-x) where G(k) = 1 + x*(2*k*x-1)/(2*k*x+x-1 - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

Extensions

Name edited by Alois P. Heinz, Dec 02 2019