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A125233
Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.
5
1, 6, 1, 15, 7, 1, 28, 22, 8, 1, 45, 50, 30, 9, 1, 66, 95, 80, 39, 10, 1, 91, 161, 175, 119, 49, 11, 1, 120, 252, 336, 294, 168, 60, 12, 1, 153, 372, 588, 630, 462, 228, 72, 13, 1, 190, 525, 960, 1218, 1092, 690, 300, 85, 14, 1, 231, 715, 1485, 2178, 2310, 1782, 990, 385, 99, 15, 1
OFFSET
0,2
COMMENTS
Left border = A000384, hexagonal numbers. The following columns are A002412, A002417, A034263, A051947, ...
Row sums = (1, 7, 23, 59, 135, 291, ...) = A126284.
A125232 is the analogous triangle for the pentagonal numbers.
REFERENCES
Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1964, p. 189.
FORMULA
T(n,0)=A000384(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>1. - R. J. Mathar, May 03 2008
EXAMPLE
First few rows of the triangle:
1;
6, 1;
15, 7, 1;
28, 22, 8, 1;
45, 50, 30, 9, 1;
66, 95, 80, 39, 10, 1;
91, 161, 175, 119, 49, 11, 1;
...
Example: (5,3) = 80 = 30 + 50 = (4,3) + (4,2).
MAPLE
A000384Psum:= proc(n, k) coeftayl( x*(1+3*x)/(1-x)^(3+k), x=0, n) ; end: A125233 := proc(n, k) A000384Psum(n-k, k) ; end: for n from 1 to 15 do for k from 0 to n -1 do printf("%d, ", A125233(n, k)) ; od: od: # R. J. Mathar, May 03 2008
MATHEMATICA
T[n_, k_] := T[n, k] = Which[k == 0, n (2 n - 1), 1 <= k < n, T[n - 1, k] + T[n - 1, k - 1], True, 0];
Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Sep 14 2023, after R. J. Mathar *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 24 2006
EXTENSIONS
Edited and extended by R. J. Mathar, May 03 2008, and M. F. Hasler, Sep 29 2012
STATUS
approved