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A240315 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k parts in which no part is unique (each part occurs at least twice). 1
1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 6, 0, 1, 0, 0, 0, 0, 0, 10, 0, 1, 0, 0, 1, 0, 7, 10, 15, 0, 1, 0, 0, 0, 1, 0, 10, 20, 21, 0, 1, 0, 0, 1, 0, 12, 1, 30, 35, 28, 0, 1, 0, 0, 0, 0, 0, 20, 0, 56, 56, 36, 0, 1, 0, 0, 1, 1, 13, 10, 126, 21, 98, 84, 45, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,26

COMMENTS

Row sums = A240085.

REFERENCES

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 87.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

Product_{i>=1} exp(x^i*y) - x^i*y = Sum_{k>=0} A_k(x)*y^k/k!.  Where A_k(x) is the o.g.f. for the number of compositions of n into k parts in which no part is unique.  In other words A_k(x) is the o.g.f. for column k.

EXAMPLE

1;

0, 0;

0, 0, 1;

0, 0, 0, 1;

0, 0, 1, 0,  1;

0, 0, 0, 0,  0,  1;

0, 0, 1, 1,  6,  0,   1;

0, 0, 0, 0,  0, 10,   0,  1;

0, 0, 1, 0,  7, 10,  15,  0,  1;

0, 0, 0, 1,  0, 10,  20, 21,  0,  1;

0, 0, 1, 0, 12,  1,  30, 35, 28,  0,  1;

0, 0, 0, 0,  0, 20,   0, 56, 56, 36,  0, 1;

0, 0, 1, 1, 13, 10, 126, 21, 98, 84, 45, 0, 1;

T(8,4) = 7 because we have: 3+3+1+1, 3+1+3+1, 3+1+1+3, 1+3+3+1, 1+3+1+3, 1+1+3+3, 2+2+2+2.

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, t!, `if`(i<1, 0,

      expand(b(n, i-1, t)+add(x^j*b(n-i*j, i-1, t+j)/j!, j=2..n/i))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):

seq(T(n), n=0..14);  # Alois P. Heinz, Apr 03 2014

MATHEMATICA

nn=10; Table[Take[Transpose[Range[0, nn]!CoefficientList[Series[ Product[Exp[x^i y]-x^i y, {i, 1, nn}], {y, 0, nn}], {y, x}]], nn+1][[j, Range[1, j]]], {j, 1, nn}]//Grid

CROSSREFS

Sequence in context: A293568 A192072 A060297 * A339431 A256041 A137378

Adjacent sequences:  A240312 A240313 A240314 * A240316 A240317 A240318

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Apr 03 2014

STATUS

approved

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Last modified January 19 09:51 EST 2021. Contains 340269 sequences. (Running on oeis4.)