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
KEYWORD
AUTHOR
Emeric Deutsch, Jan 29 2016
STATUS
approved