A361475 -id:A361475 - cp's OEIS frontend

A361476 Antidiagonal sums of A361475.

0, 1, 4, 12, 34, 99, 308, 1040, 3820, 15197, 65060, 297828, 1449742, 7468527, 40555732, 231335944, 1381989864, 8623700793, 56078446596, 379233142780, 2662013133274, 19362917621979, 145719550012276, 1133023004941248, 9090156910550084, 75161929739797493, 639793220877941476

Offset: 0

Stefano Spezia, Mar 13 2023

nonn

Robert Israel, Table of n, a(n) for n = 0..596

Cf. A026898, A104878, A361475.

f:= proc(n) local k;
add( (k^(n-k+2) - 1)/(k - 1),k=2..n+2)
end proc:
map(f, [$0..30]); # Robert Israel, Nov 12 2024

A361475[n_,k_]:=(k^n-1)/(k-1); a[n_]:=Sum[A361475[n-k+2,k],{k,2,n+2}]; Array[a,27,0]

a(n) = Sum_{k=2..n+2} (k^(n-k+2) - 1)/(k - 1).
a(n) ~ A026898(n).
a(n) = Sum_{k=0..n} k * A104878(n,k). - Alois P. Heinz, Dec 05 2023

A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 9, 2, 0, 49, 32, 3, 0, 225, 338, 75, 4, 0, 961, 3200, 1323, 144, 5, 0, 3969, 29282, 21675, 3844, 245, 6, 0, 16129, 264992, 348843, 97344, 9245, 384, 7, 0, 65025, 2389298, 5589675, 2439844, 335405, 19494, 567, 8, 0, 261121, 21516800, 89467563, 61027344, 12090125, 960000, 37303, 800, 9, 0

Offset: 0

Stefano Spezia, Dec 26 2024

			The array begins as:
    0,     0,      0,       0,        0,        0, ...
    1,     2,      3,       4,        5,        6, ...
    9,    32,     75,     144,      245,      384, ...
   49,   338,   1323,    3844,     9245,    19494, ...
  225,  3200,  21675,   97344,   335405,   960000, ...
  961, 29282, 348843, 2439844, 12090125, 47073606, ...
  ...

Cf. A027620, A060867 (k=2), A060868 (k=3), A060869 (k=4), A060870 (k=5), A060871 (k=7), A361475, A379588 (antidiagonal sums).

A[n_,k_]:=(k^n-1)^2/(k-1); Table[A[n-k+2,k],{n,0,9},{k,2,n+2}]//Flatten

G.f. of column k: (1 - k)*x*(1 + k*x)/((1 - x)*(1 - k*x)*(1 - k^2*x)).
E.g.f. of column k: exp(x)*(1 - 2*exp((k-1)*x) + exp((k^2-1)*x))/(k - 1).
A(2, n) = A027620(n-2) for n > 1.

cp's OEIS Frontend

A361476 Antidiagonal sums of A361475.

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