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A117427 Triangle T, read by rows, equal to the matrix square of triangle A117418; also equals a column bisection of triangle A117418: column 2k+1 of T^(1/2) equals column k of T. 4
1, 2, 1, 4, 4, 1, 9, 14, 6, 1, 23, 50, 28, 8, 1, 66, 191, 126, 48, 10, 1, 209, 780, 572, 264, 70, 12, 1, 724, 3415, 2692, 1440, 438, 98, 14, 1, 2722, 15924, 13203, 7990, 2646, 698, 130, 16, 1, 11054, 78595, 67542, 45566, 15920, 4702, 1029, 168, 18, 1, 48221, 408337, 360117, 268072, 96848, 31030, 7536, 1472, 206, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
FORMULA
T(n, k) = A117418(n+k+1, 2*k+1). - G. C. Greubel, May 31 2021
EXAMPLE
Triangle T begins:
1;
2, 1;
4, 4, 1;
9, 14, 6, 1;
23, 50, 28, 8, 1;
66, 191, 126, 48, 10, 1;
209, 780, 572, 264, 70, 12, 1;
724, 3415, 2692, 1440, 438, 98, 14, 1;
2722, 15924, 13203, 7990, 2646, 698, 130, 16, 1;
11054, 78595, 67542, 45566, 15920, 4702, 1029, 168, 18, 1;
The matrix square-root T^(1/2) = A117418, which begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 8, 4, 1;
1, 23, 22, 14, 5, 1;
1, 66, 65, 50, 20, 6, 1;
1, 209, 208, 191, 79, 28, 7, 1;
where column 2k+1 of T^(1/2) equals column k of T for k>=0.
MATHEMATICA
A117418[n_, k_]:= A117418[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==n-1, n, Sum[A117418[n-Floor[(k+1)/2], Floor[k/2] +j]*A117418[Floor[(k-1)/2] +j, Floor[(k-1)/2]], {j, 0, n-k}] ]]];
A117427[n_, k_]:= A117418[n+k+1, 2*k+1];
Table[A117427[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)
PROG
(Sage)
@CachedFunction
def A117418(n, k):
if (k<0 or k>n): return 0
elif (k==0 or k==n): return 1
elif (k==n-1): return n
else: return sum( A117418(n -(k+1)//2, k//2 +j)*A117418((k-1)//2 +j, (k-1)//2) for j in (0..n-k))
def A117427(n, k): return A117418(n+k+1, 2*k+1)
flatten([[A117427(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021
CROSSREFS
Cf. A117418 (T^(1/2)), A117425 (dual), A117428 (row sums).
Sequence in context: A263989 A202710 A200965 * A097761 A200756 A108556
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 14 2006
STATUS
approved

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Last modified April 15 10:24 EDT 2024. Contains 371681 sequences. (Running on oeis4.)