OFFSET
1,5
COMMENTS
Row sums are: {1, 2, 12, 130, 2030, 48642, 1882762, 121744898, 13337520498, 2503662940162, ...}.
REFERENCES
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 176
LINKS
G. C. Greubel, Rows n = 1..75 of triangle, flattened
FORMULA
T(n,k) = T(n-1, k-1) + q^k*T(n-1, k), with q=3.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 10, 1;
1, 91, 37, 1;
1, 820, 1090, 118, 1;
1, 7381, 30250, 10648, 361, 1;
1, 66430, 824131, 892738, 98371, 1090, 1;
1, 597871, 22317967, 73135909, 24796891, 892981, 3277, 1;
MAPLE
T:= proc(n, k) option remember;
q:=3;
if k=1 or k=n then 1
else T(n-1, k-1) + q^k*T(n-1, k)
fi; end:
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 22 2019
MATHEMATICA
q:=3; T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, q^k*T[n-1, k] + T[n-1, k-1]]; Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Nov 22 2019 *)
PROG
(PARI) T(n, k) = my(q=3); if(k==1 || k==n, 1, q^k*T(n-1, k) + T(n-1, k-1)); \\ G. C. Greubel, Nov 22 2019
(Magma)
function T(n, k)
q:=3;
if k eq 1 or k eq n then return 1;
else return T(n-1, k-1) + q^k*T(n-1, k);
end if; return T; end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 22 2019
(Sage)
@CachedFunction
def T(n, k):
q=3;
if (k==1 or k==n): return 1
else: return q^k*T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 22 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 12 2010
EXTENSIONS
Edited by G. C. Greubel, Nov 22 2019
STATUS
approved