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A166962 Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n. 3
1, 1, 1, 1, 12, 1, 1, 103, 69, 1, 1, 834, 2170, 316, 1, 1, 6685, 53910, 27830, 1329, 1, 1, 53496, 1219015, 1652300, 281195, 5412, 1, 1, 427987, 26455251, 81939195, 34800675, 2487917, 21781, 1, 1, 3423918, 563692024, 3700851816, 3327253410 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Row sums are 1, 2, 14, 174, 3322, 89756, 3211420, 146132808, 8202467544, 554489060200,..
The recursion relation T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) connects several sequences for differing values of m. These are: m = 0 yields A008277, m = 1 yields A166960, m = 2 yields A166961, and m = 3 yields this sequence. These sequences are, in essence, a variation of Stirling numbers of the second kind. - G. C. Greubel, May 29 2016
LINKS
FORMULA
T(n, k) = (3*n - 3*k + 1)*T(n - 1, k - 1) + k*(3*k - 2)*T(n - 1, k). - G. C. Greubel, May 29 2016
EXAMPLE
1;
1, 1;
1, 12, 1;
1, 103, 69, 1;
1, 834, 2170, 316, 1;
1, 6685, 53910, 27830, 1329, 1;
1, 53496, 1219015, 1652300, 281195, 5412, 1;
1, 427987, 26455251, 81939195, 34800675, 2487917, 21781, 1;
1, 3423918, 563692024, 3700851816, 3327253410, 586846782, 20312292, 87300, 1;
MAPLE
A166962 := proc(n, k)
if k = 1 or k = n then
1;
elif n <= 2 then
1;
else
(3*n-3*k+1)*procname(n-1, k-1)+k*(3*k-2)*procname(n-1, k) ;
end if;
end proc: # R. J. Mathar, Mar 26 2013
MATHEMATICA
A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := (3*n - 3*k + 1)*A[n - 1, k - 1] + k*(3*k - 2)*A[n - 1, k]; Flatten[Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)
CROSSREFS
Sequence in context: A155491 A142460 A156280 * A022175 A340427 A176627
KEYWORD
nonn,easy,tabl
AUTHOR
Roger L. Bagula and Mats Granvik, Oct 25 2009
STATUS
approved

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Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)