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A263232 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 3 (n >= 0, 0 <= k <= floor(n/3)). 3
1, 1, 2, 2, 1, 4, 1, 5, 2, 8, 2, 1, 10, 4, 1, 15, 5, 2, 19, 8, 2, 1, 27, 10, 4, 1, 34, 15, 5, 2, 47, 19, 8, 2, 1, 59, 27, 10, 4, 1, 79, 34, 15, 5, 2, 99, 47, 19, 8, 2, 1, 130, 59, 27, 10, 4, 1, 162, 79, 34, 15, 5, 2, 209, 99, 47, 19, 8, 2, 1, 259, 130, 59, 27, 10, 4, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row n has 1+floor(n/3) terms. Row sums are the partition numbers (A000041). T(n,0) = A027337(n). Sum_{k=0..floor(n/3)} k*T(n,k) = A024787(n).

LINKS

Alois P. Heinz, Rows n = 0..300, flattened

FORMULA

G.f.: (1-x)*(1-x^2)*(1-tx^3)*Product_{j>=4} (1-x^j).

EXAMPLE

T(7,1) = 4 because we have [4,3], [3,2,2], [3,2,1,1], and [3,1,1,1,1].

T(9,2) = 2 because we have [3,3,2,1] and [3,3,1,1,1].

Triangle starts:

  1;

  1;

  2;

  2, 1;

  4, 1;

  5, 2;

  8, 2, 1.

MAPLE

g := 1/((1-x)*(1-x^2)*(1-t*x^3)*(product(1-x^j, j = 4 .. 80))): gser := simplify(series(g, x = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n, i) option remember; expand(

      `if`(n=0, 1, `if`(i<1, 0, `if`(i=3, x, 1)*

      `if`(i>n, 0, b(n-i, i)) +b(n, i-1))))

    end:

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

seq(T(n), n=0..25);  # Alois P. Heinz, Nov 01 2015

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, If[i == 3, x, 1]* If[i > n, 0, b[n - i, i]] + b[n, i - 1]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-Fran├žois Alcover, Jan 21 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A027337, A024787, A116599.

Sequence in context: A018219 A174714 A116633 * A134666 A129712 A051720

Adjacent sequences:  A263229 A263230 A263231 * A263233 A263234 A263235

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Nov 01 2015

STATUS

approved

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Last modified November 14 04:40 EST 2019. Contains 329108 sequences. (Running on oeis4.)