A123491
Diagonal sums of number triangle A123490.
Original entry on oeis.org
1, 2, 5, 10, 22, 48, 112, 274, 715, 1982, 5837, 18180, 59644, 205296, 739032, 2775180, 10846965, 44039754, 185391469, 807776198, 3637193474, 16900721824, 80939650552, 399061251246, 2023408865983, 10540656630118
Offset: 0
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Table[Sum[((k + 2)^(n - 2 k) + k)/(k + 1), {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Oct 14 2017 *)
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for(n=0,25, print1(sum(k=0,floor(n/2), ((k + 2)^(n - 2 k) + k)/(k + 1)), ", ")) \\ G. C. Greubel, Oct 14 2017
A103439
a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.
Original entry on oeis.org
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
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[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
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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
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Join[{0},Table[Sum[Sum[(i-j+1)^j,{j,0,i}],{i,0,n}],{n,0,30}]] (* Harvey P. Dale, Dec 03 2018 *)
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a(n) = sum(i=0, n-1, sum(j=0, i, (i-j+1)^j)); \\ Michel Marcus, Jun 15 2021
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[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
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