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A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k). 36
1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 166, 546, 166, 1, 1, 677, 5482, 5482, 677, 1, 1, 2724, 47175, 109640, 47175, 2724, 1, 1, 10915, 373809, 1709675, 1709675, 373809, 10915, 1, 1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Consider the triangle T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k). For m = ...,-2,-1,0,1,2,3,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, ... - _N. J. A. Sloane_, May 08 2013
LINKS
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_3(n,k).
FORMULA
T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 3.
Sum_{k=1..n} T(n, k) = A008544(n-1).
From _G. C. Greubel_, Mar 14 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = A144414(n-1).
T(n, 3) = A142976(n-2).
T(n, 4) = A144380(n-3).
T(n, 5) = A144381(n-4). (End)
EXAMPLE
The rows n >= 1 and columns 1 <= k <= n look as follows:
1;
1, 1;
1, 8, 1;
1, 39, 39, 1;
1, 166, 546, 166, 1;
1, 677, 5482, 5482, 677, 1;
1, 2724, 47175, 109640, 47175, 2724, 1;
1, 10915, 373809, 1709675, 1709675, 373809, 10915, 1;
1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1;
MAPLE
A142458 := proc(n, k) if n = k then 1; elif k > n or k < 1 then 0 ; else (3*n-3*k+1)*procname(n-1, k-1)+(3*k-2)*procname(n-1, k) ; end if; end proc:
seq(seq(A142458(n, k), k=1..n), n=1..10) ; # _R. J. Mathar_, Jun 04 2011
MATHEMATICA
T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m] ];
Table[T[n, k, 3], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by _G. C. Greubel_, Mar 14 2022 *)
PROG
(Sage)
def T(n, k, m): # A142458
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
flatten([[T(n, k, 3) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Mar 14 2022
CROSSREFS
Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), this sequence (m=3), A142459 (m=4), A142560 (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).
Sequence in context: A152972 A166346 A157640 * A174528 A259465 A176227
KEYWORD
nonn,easy,tabl
AUTHOR
_Roger L. Bagula_, Sep 19 2008
EXTENSIONS
Edited by the Associate Editors of the OEIS, Aug 28 2009
STATUS
approved

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Last modified June 17 15:50 EDT 2024. Contains 373461 sequences. (Running on oeis4.)