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Search: id:A097364
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| A097364 |
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Triangle read by rows, 0<=k<n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k. |
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4
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| 1, 2, 0, 2, 1, 0, 3, 1, 1, 0, 2, 3, 1, 1, 0, 4, 2, 3, 1, 1, 0, 2, 5, 3, 3, 1, 1, 0, 4, 4, 6, 3, 3, 1, 1, 0, 3, 6, 6, 7, 3, 3, 1, 1, 0, 4, 6, 10, 7, 7, 3, 3, 1, 1, 0, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 0, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 0, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 0, 4, 10, 21, 22, 24, 17
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OFFSET
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1,2
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COMMENT
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Sum(T(n,k): 0<=k<n) = A000041(n); T(n,0)+T(n,1) = n for n>1;
T(n,0) = A000005(n); T(n,1) = A049820(n) for n>1;
T(n,2) = floor((n-2)/2))*(floor((n-2)/2))+1)/2 = A000217(floor((n-2)/2))) = A008805(n-4) for n>3;
Without the 0's (which are of no consequence for the triangle) it is A116685. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2006
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FORMULA
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G.f.=sum(x^i/(1-x^i)/product(1-tx^j,j=1..i-1), i=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2006
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EXAMPLE
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T(8,0)=4: 8=4+4=2+2+2+2=1+1+1+1+1+1+1+1,
T(8,1)=4: 3+3+2=2+2+2+1+1=2+2+1+1+1+1=2+1+1+1+1+1+1,
T(8,2)=6: 5+3=4+2+2=3+3+1+1=3+2+2+1=3+2+1+1+1=3+1+1+1+1+1,
T(8,3)=3: 4+3+1=4+2+1+1=4+1+1+1+1,
T(8,4)=3: 6+2=5+2+1=5+1+1+1,
T(8,5)=1: 6+1+1,
T(8,6)=1: 7+1,
T(8,7)=0;
Sum(T(8,k): 0<=k<8) = 4+4+6+3+3+1+1+0 = 22 = A000041(8).
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MAPLE
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g:=sum(x^i/(1-x^i)/product(1-t*x^j, j=1..i-1), i=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 15 do P[n]:=coeff(gser, x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2006
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CROSSREFS
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Cf. A116685.
Adjacent sequences: A097361 A097362 A097363 this_sequence A097365 A097366 A097367
Sequence in context: A133696 A127371 A036849 this_sequence A074905 A058548 A080844
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KEYWORD
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nonn,tabl
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 09 2004
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