A394590 Number of permutations of [2*n] with the number moved left exceeding the number moved right by n.

1, 0, 1, 6, 267, 5050, 520598, 20713252, 3891019939, 266156640474, 79482588671154, 8325744599377260, 3624674791200639406, 539424704140269574276, 322575897023940844552204, 64712722269670777415835976, 50899350828638565957558873763, 13240165399882959143766305222394

Offset

0,4

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = A062866(2*n,n) = Sum_{e=0..floor(n/2)} binomial(2*n,n+2*e) * A271697(n+2*e,e).

Example

a(3) = 6 because the 6 permutations of [6] where the number of elements moved left exceeds the number of elements moved right by 3 are:
  [1, 6, 2, 3, 4, 5] (Fixed: {1}, Right: {2->6}, Left: {3->2, 4->3, 5->4, 6->5})
  [6, 2, 1, 3, 4, 5] (Fixed: {2}, Right: {1->6}, Left: {3->1, 4->3, 5->4, 6->5})
  [6, 1, 3, 2, 4, 5] (Fixed: {3}, Right: {1->6}, Left: {2->1, 4->2, 5->4, 6->5})
  [6, 1, 2, 4, 3, 5] (Fixed: {4}, Right: {1->6}, Left: {2->1, 3->2, 5->3, 6->5})
  [6, 1, 2, 3, 5, 4] (Fixed: {5}, Right: {1->6}, Left: {2->1, 3->2, 4->3, 6->4})
  [5, 1, 2, 3, 4, 6] (Fixed: {6}, Right: {1->5}, Left: {2->1, 3->2, 4->3, 5->4}).

Prog

(PARI) E(n, k) = sum(j=0, k, (-1)^j*binomial(n+1, j)*(k+1-j)^n);
a271697(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n, j)*E(j, k));
a(n) = sum(e=0, n\2, binomial(2*n, n+2*e)*a271697(n+2*e, e));

Crossrefs

Cf. A062866, A271697.

Sequence in context: A221900 A383881 A229773 * A381365 A193983 A348701

Adjacent sequences: A394587 A394588 A394589 * A394591 A394592 A394593

Keyword

nonn

Author

Seiichi Manyama, Apr 23 2026

Status

approved