A327710 -id:A327710 - cp's OEIS frontend

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

Emeric Deutsch, Jan 29 2016

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.

			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;
  ...

Alois P. Heinz, Rows n = 1..1000, flattened
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

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

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

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 *)

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

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

A328222 Number of compositions of n into distinct parts such that the difference between adjacent parts is at least two.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 7, 11, 15, 23, 31, 47, 63, 91, 133, 183, 255, 355, 487, 661, 973, 1287, 1783, 2411, 3289, 4383, 5905, 8337, 10975, 14829, 19783, 26451, 34945, 46403, 60539, 84109, 108471, 144591, 189013, 250257, 324451, 426705, 550443, 715421, 961249

Offset: 0

Alois P. Heinz, Feb 24 2020

nonn

All terms are odd.

			a(7) = 7: 142, 241, 25, 52, 16, 61, 7.
a(8) = 11: 314, 413, 152, 251, 35, 53, 26, 62, 17, 71, 8.
a(9) = 15: 135, 153, 315, 351, 513, 531, 162, 261, 36, 63, 27, 72, 18, 81, 9.
a(10) = 23: 2413, 3142, 253, 352, 415, 514, 136, 163, 316, 361, 613, 631, 46, 64, 172, 271, 37, 73, 28, 82, 19, 91, 10.

Cf. A002464, A003114, A003242, A032020, A327710, A332829.

cp's OEIS Frontend

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.

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