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A125234
Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.
3
1, 7, 1, 18, 8, 1, 34, 26, 9, 1, 55, 60, 35, 10, 1, 81, 115, 95, 45, 11, 1, 112, 196, 210, 140, 56, 12, 1, 148, 308, 406, 350, 196, 68, 13, 1, 189, 456, 714, 756, 546, 264, 81, 14, 1, 235, 645, 1170, 1470, 1302, 810, 345, 95, 15, 1, 286, 880, 1815, 2640, 2772, 2112, 1155, 440, 110, 16, 1
OFFSET
1,2
COMMENTS
The leftmost column contains the heptagonal numbers A000566.
The adjacent columns to the right are A002413, A002418, A027800, A051946, A050484.
Row sums = 1, 8, 27, 70, 161, 348, 727, ... = 6*(2^n-1)-5*n.
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1966, p. 189.
FORMULA
T(n,0) = A000566(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>0.
EXAMPLE
First few rows of the triangle are:
1;
7, 1;
18, 8, 1;
34, 26, 9, 1;
55, 60, 35, 10, 1;
81, 115, 95, 45, 11, 1;
112, 196, 210, 140, 56, 12, 1;
Example: T(6,2) = 95 = 35 + 60 = T(5,2) + T(5,1).
MAPLE
A000566 := proc(n) n*(5*n-3)/2 ; end: A125234 := proc(n, k) if k = 0 then A000566(n); elif k>= n then 0 ; else procname(n-1, k-1)+procname(n-1, k) ; fi; end: seq(seq(A125234(n, k), k=0..n-1), n=1..16) ; # R. J. Mathar, Sep 09 2009
MATHEMATICA
A000566[n_] := n(5n-3)/2;
T[n_, k_] := Which[k == 0, A000566[n], k >= n, 0, True, T[n-1, k-1] + T[n-1, k] ];
Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)
CROSSREFS
Analogous triangles for the hexagonal and pentagonal numbers are A125233 and A125232.
Sequence in context: A013614 A179530 A098081 * A028325 A245484 A215503
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 24 2006
EXTENSIONS
Edited and extended by R. J. Mathar, Sep 09 2009
STATUS
approved