A268187 Triangle read by rows: T(n,k) is the number of no-leg partitions of n having Durfee square of size k (n >= 1, 1 <= k <= floor(sqrt(n))). Also, number of no-arm partitions of n having Durfee square of size k.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 1, 4, 2, 1, 5, 3, 1, 5, 4, 1, 6, 5, 1, 6, 7, 1, 7, 8, 1, 1, 7, 10, 1, 1, 8, 12, 2, 1, 8, 14, 3, 1, 9, 16, 5, 1, 9, 19, 6, 1, 10, 21, 9, 1, 10, 24, 11, 1, 11, 27, 15, 1, 11, 30, 18, 1, 1, 12, 33, 23, 1

Offset

1,9

Comments

Given a partition P, the partition formed by the cells situated below the Durfee square of P is called the leg of P. Similarly, the partition formed by the cells situated to the right of the Durfee square of P is called the arm of P.

Links

Alois P. Heinz, Rows n = 1..1000, flattened

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Formula

G.f.: G(t,x) = Sum_{k>=0} ( t^k*x^(k^2)/Product_{i=1..k} (1-x^i) ).

Sum_{k>=0} T(n,k) = A003114(n).

Sum_{k>=1} k * T(n,k) = A268188(n).

Sum_{k>=0} k! * T(n,k) = A327710(n). - Alois P. Heinz, Feb 25 2020

Example

T(9,2) = 3 because we have [7,2], [6,3], and [5,4].
Triangle begins:
  1;
  1;
  1;
  1, 1;
  1, 1;
  1, 2;
  1, 2;
  1, 3;
  1, 3, 1;
  1, 4, 1;
  1, 4, 2;
  1, 5, 3;
  1, 5, 4;
  1, 6, 5;
  1, 6, 7;
  1, 7, 8, 1;
  ...

Maple

G := add(t^k*x^(k^2)/mul(1-x^i, i = 1 .. k), k = 0 .. 80): Gser := simplify(series(G, x = 0, 40)): for n to 35 do P[n] := sort(coeff(Gser, x, n)) end do: for n to 35 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> b(n-k^2, k):
seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Jan 30 2016

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := b[n-k^2, k]; Table[T[n, k], {n, 1, 30}, {k, 1, Floor[Sqrt[n]]}] // Flatten (* Jean-François Alcover, Dec 10 2016 after Alois P. Heinz *)

Prog

(PARI) T(n, k) = polcoef(1/prod(j=1, k, 1-x^j+x*O(x^n)), n-k*k);
tabf(nn) = for(n=1, nn, for(k=1, sqrtint(n), print1(T(n, k), ", ")); print) \\ Seiichi Manyama, Oct 14 2019

Crossrefs

Cf. A003114, A115994, A268188, A327710, A328346.

Sequence in context: A137163 A344233 A072625 * A232440 A355749 A278538

Adjacent sequences: A268184 A268185 A268186 * A268188 A268189 A268190

Keyword

nonn,tabf,look

Author

Emeric Deutsch, Jan 29 2016

Status

approved