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A209669 T(n,k) = count of degree k monomials in the elementary symmetric polynomials e(mu,k) summed over all partitions mu of n. 6
1, 1, 5, 1, 10, 37, 1, 21, 120, 405, 1, 42, 363, 1644, 5251, 1, 85, 1117, 6814, 27405, 84893, 1, 170, 3360, 27404, 138085, 514248, 1556535, 1, 341, 10164, 111045, 701960, 3145848, 11133493, 33175957, 1, 682, 30520, 445132, 3521405, 18956548, 78337448 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,2) = A000975(n+1) because only partitions into parts of 1 and 2 contribute, so that T(n,2) = Sum_{k=0..floor(n/2)} 2^(n-2k). - Peter J. Taylor, Mar 01 2017

LINKS

Peter J. Taylor, Table of n, a(n) for n = 1..5050

Peter J. Taylor, Python program to compute terms

Wikipedia, Symmetric polynomials

FORMULA

T(n,k) = Sum_{lambda} Product_{i} binomial(k, lambda_i) where the sum is over partitions of n. - Peter J. Taylor, Mar 01 2017

EXAMPLE

Table starts as

1;

1,  5;

1, 10,  37;

1, 21, 120,  405;

1, 42, 363, 1644, 5251;

...

For n = 2, k = 2 the partitions of n are 2 and 1+1, which correspond respectively to (xy) contributing 1 and (x+y)*(x+y) contributing 4 for a total of 5. - Peter J. Taylor, Mar 01 2017

MATHEMATICA

e[n_, v_] := Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#] == n &]]; e[par_?PartitionQ, v_] := Times @@ (e[#, v] & /@ par); Table[Tr[(e[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

PROG

(Python) See Taylor link

CROSSREFS

Row sums are A209670, main diagonal is A209671.

Second column is A000975 offset by 1. - Peter J. Taylor, Mar 01 2017

Sequence in context: A116547 A013612 A112830 * A189745 A062967 A245211

Adjacent sequences:  A209666 A209667 A209668 * A209670 A209671 A209672

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Mar 11 2012

STATUS

approved

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Last modified October 17 14:47 EDT 2019. Contains 328114 sequences. (Running on oeis4.)