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A154229 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1), read by rows. 6
1, 1, 1, 1, 38, 1, 1, 139, 139, 1, 1, 365, 8828, 365, 1, 1, 807, 70492, 70492, 807, 1, 1, 1592, 357459, 7062136, 357459, 1592, 1, 1, 2889, 1404923, 98777227, 98777227, 1404923, 2889, 1, 1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 40, 280, 9560, 142600, 7780240, 200370080, 16155726160, ...}.
The row sums of this class of sequences (see cross-references) 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)^2 = A000537(n+1). - G. C. Greubel, Mar 02 2021
LINKS
FORMULA
T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 38, 1;
1, 139, 139, 1;
1, 365, 8828, 365, 1;
1, 807, 70492, 70492, 807, 1;
1, 1592, 357459, 7062136, 357459, 1592, 1;
1, 2889, 1404923, 98777227, 98777227, 1404923, 2889, 1;
1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 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)^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]^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)^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)^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
Cf. A000537.
Sequence in context: A023930 A022072 A351221 * A225433 A225398 A037936
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 05 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 02 2021
STATUS
approved

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Last modified July 13 12:04 EDT 2024. Contains 374282 sequences. (Running on oeis4.)