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A154227
Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
6
1, 1, 1, 1, 8, 1, 1, 19, 19, 1, 1, 35, 158, 35, 1, 1, 57, 592, 592, 57, 1, 1, 86, 1629, 5608, 1629, 86, 1, 1, 123, 3767, 28549, 28549, 3767, 123, 1, 1, 169, 7760, 105621, 309458, 105621, 7760, 169, 1, 1, 225, 14694, 320566, 1985274, 1985274, 320566, 14694, 225, 1
OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 10, 40, 230, 1300, 9040, 64880, 536560, 4641520, ...}.
The row sums of this class of sequences (see Cf section) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = binomial(n+2, 2). - G. C. Greubel, Mar 02 2021
FORMULA
T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 8, 1;
1, 19, 19, 1;
1, 35, 158, 35, 1;
1, 57, 592, 592, 57, 1;
1, 86, 1629, 5608, 1629, 86, 1;
1, 123, 3767, 28549, 28549, 3767, 123, 1;
1, 169, 7760, 105621, 309458, 105621, 7760, 169, 1;
1, 225, 14694, 320566, 1985274, 1985274, 320566, 14694, 225, 1;
MAPLE
T:= proc(n, k) option remember;
if k=0 or k=n then 1
else T(n-1, k) + T(n-1, k-1) + binomial(n+2, 2)*T(n-2, k-1)
fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + Binomial[n+2, 2]*T[n-2, k-1] ];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)
PROG
(Sage)
def f(n): return binomial(n+2, 2)
def T(n, k):
if (k==0 or k==n): return 1
else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
f:= func< n | Binomial(n+2, 2) >;
function T(n, k)
if k eq 0 or k eq n then return 1;
else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 05 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 02 2021
STATUS
approved