OFFSET
0,4
COMMENTS
Motivated by a proposal from Charlie Marion.
REFERENCES
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd ed.; Addison-Wesley, 1994, pp. 283-290.
FORMULA
A(k, n) = Sum_{j=0..k} (n+j)^2, for k >= 0, n >= 0.
A(k, n) = Sum_{j=0..n+k} j^2 - (2*n-1)*n*(n-1)/3! = S(n+k) - (2*n-1)*n*(n-1)/3!, with S(n+k) = (1/3)*Sum_{j=0..2} binomial(3, j)*B_j*(n+k+1)^(3-j), with the Bernoulli numbers A027641 / A027642 (see Graham et al., pp. 283-290).
Recurrence for sequence of row k: A(k, n) = A(k, n-1) + (k+1)*(2*n + k - 1), n >= 1, with A(k, 0) = (2*k+1)*(k+1)*k/3!, for k >= 0.
EXAMPLE
The array A(k, n) begins:
k \ n 0 1 2 3 4 5 6 7 8 9 10 ...
-----------------------------------------------------------
0: 0 1 4 9 16 25 36 49 64 81 100 ...
1: 1 5 13 25 41 61 85 113 145 181 221 ...
2: 5 14 29 50 77 110 149 194 245 302 365 ...
3: 14 30 54 86 126 174 230 294 366 446 534 ...
4: 30 55 90 135 190 255 330 415 510 615 730 ...
5: 55 91 139 199 271 355 451 559 679 811 955 ...
6: 91 140 203 280 371 476 595 728 875 1036 1211 ...
7: 140 204 284 380 492 620 764 924 1100 1292 1500 ...
8: 204 285 384 501 636 789 960 1149 1356 1581 1824 ...
9: 285 385 505 645 805 985 1185 1405 1645 1905 2185 ...
...
-----------------------------------------------------------
The triangle T(m, n) begins:
m \ n 0 1 2 3 4 5 6 7 8 9 ...
-----------------------------------------------------------
0: 0
1: 1 1
2: 5 5 4
3: 14 14 13 9
4: 30 30 29 25 16
5: 55 55 54 50 41 25
6: 91 91 90 86 77 61 36
7: 140 140 139 135 126 110 85 49
8: 204 204 203 199 190 174 149 113 64
9: 285 285 284 280 271 255 230 194 145 81
...
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CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 27 2021
STATUS
approved