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A160870
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Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).
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34
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1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 7, 13, 15, 1, 1, 6, 35, 40, 31, 1, 1, 12, 31, 155, 121, 63, 1, 1, 8, 91, 156, 651, 364, 127, 1, 1, 15, 57, 600, 781, 2667, 1093, 255, 1, 1, 13, 155, 400, 3751, 3906, 10795, 3280, 511, 1, 1, 18, 130, 1395, 2801, 22932, 19531, 43435, 9841, 1023, 1
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OFFSET
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1,5
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REFERENCES
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Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]
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LINKS
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FORMULA
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T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1).
T(n,k) = Sum_{d|n} (n/d)^(k-1) * T(d, k-1).
T(Product(p^e), k) = Product(Gaussian_poly[e+k-1, e]_p).
(End)
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EXAMPLE
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Array begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,...
1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,...
1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,...
1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,...
...
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MATHEMATICA
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T[_, 1] = 1; T[1, _] = 1; T[n_, k_] := T[n, k] = DivisorSum[n, (n/#)^(k-1) *T[#, k-1]&]; Table[T[n-k+1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2015 *)
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PROG
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(PARI):
adu(M)=
{ /* Read by AntiDiagonals, Upwards */
local(N=matsize(M)[1]);
for (n=1, N, for(j=0, n-1, print1(M[n-j, j+1], ", ") ) );
}
T(n, k)=
{
if ( (n==1) || (k==1), return(1) );
return( sumdiv(n, d, d*T(d, k-1)) );
}
M=matrix(15, 15, n, k, T(n, k)) /* square array */
adu(M) /* sequence */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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