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A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k even parts (n >= 0, k >= 0). 2
1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 3, 1, 2, 4, 2, 2, 5, 3, 2, 6, 4, 3, 7, 4, 1, 3, 8, 6, 1, 3, 10, 8, 1, 4, 11, 10, 2, 5, 13, 11, 3, 5, 15, 14, 4, 5, 18, 18, 5, 6, 20, 21, 7, 7, 23, 24, 9, 1, 8, 26, 29, 12, 1, 8, 30, 36, 14, 1, 9, 34, 41, 18, 2, 11, 38, 47, 23, 3, 12, 43, 55, 28, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
COMMENTS
Row n contains floor((1 + sqrt(1+4*n))/2) terms.
Row sums yield A000009.
T(n,0) = A000700(n), T(n,1) = A096911(n) for n >= 1.
Sum_{k>=0} k*T(n,k) = A116680(n).
LINKS
FORMULA
G.f.: Product_{j>=1} (1+x^(2*j-1))*(1+t*x^(2*j)).
EXAMPLE
T(9,2)=2 because we have [6,2,1] and [4,3,2].
Triangle starts:
1;
1;
0, 1;
1, 1;
1, 1;
1, 2;
1, 2, 1;
1, 3, 1;
MAPLE
g:=product((1+x^(2*j-1))*(1+t*x^(2*j)), j=1..25): gser:=simplify(series(g, x=0, 38)): P[0]:=1: for n from 1 to 27 do P[n]:=sort(coeff(gser, x^n)) od: for n from 0 to 27 do seq(coeff(P[n], t, j), j=0..floor((sqrt(1+4*n)-1)/2)) od; # yields sequence in triangular form
MATHEMATICA
With[{m=25}, CoefficientList[CoefficientList[Series[Product[(1+x^(2*j- 1))*(1+t*x^(2*j)), {j, 1, m+2}], {x, 0, m}, {t, 0, m}], x], t]]//Flatten (* G. C. Greubel, Jun 07 2019 *)
CROSSREFS
Sequence in context: A008289 A326625 A188884 * A350032 A146290 A347045
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 22 2006
STATUS
approved

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Last modified July 15 07:28 EDT 2024. Contains 374324 sequences. (Running on oeis4.)