A135757 -id:A135757 - cp's OEIS frontend

A135758 Catalan numbers at triangular positions: a(n) = A000108(n(n+1)/2).

1, 1, 5, 132, 16796, 9694845, 24466267020, 263747951750360, 11959798385860453492, 2257117854077248073253720, 1759414616608818870992479875972, 5632681584560312734993915705849145100

Offset: 0

Paul D. Hanna, Dec 02 2007

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A000108, A135757.

With[{nn=100},Join[{1},Pick[CatalanNumber[Range[nn]],Array[ IntegerQ[ (Sqrt[1+8#]-1)/2]&,nn]]]] (* Harvey P. Dale, Sep 24 2011 *)
Table[Binomial[n*(n + 1), n*(n + 1)/2]/(1 + Binomial[n + 1, 2]), {n,0,10}] (* G. C. Greubel, Nov 07 2016 *)

a(n)=binomial(n*(n+1),n*(n+1)/2)/(n*(n+1)/2+1)

a(n) = C(n(n+1), n(n+1)/2) / [n(n+1)/2 + 1].

A109901 a(n) = binomial(n^2, n*(n+1)/2).

Original entry on oeis.org

1, 1, 4, 84, 8008, 3268760, 5567902560, 39049918716424, 1118770292985239888, 130276394656770614583240, 61448471214136179596720592960, 117118180539414377821494470432491764, 900390992257782351906806257139068209113040, 27883369051325994219981405855549095749234579210080

Offset: 0

Amarnath Murthy, Jul 14 2005

8*a(2*n+1)^4 = A182010(n) = number of potential group developed cocyclic Hadamard matrices over (the group) Z_{(2*n+1)^2} X Z^2_2 [Baliga, et al., p. 130]. - L. Edson Jeffery, Apr 10 2012

			a(6) = 36!/(21!*15!) = 5567902560.

Andrew Howroyd, Table of n, a(n) for n = 0..50
A. Baliga and K. J. Horadam, Cocyclic Hadamard matrices over Z_t X Z^2_2, Australas. J. Combin. 11(1995), 123-134.

Cf. variants: A014062 (C(n^2,n*(n-1))), A135757 (C(n*(n+1),n*(n+1)/2)).

Cf. A182010.

seq(binomial(n^2,n*(n+1)/2),n=0..12); # Emeric Deutsch, Jul 16 2005

Table[Binomial[n^2,(n(n+1))/2],{n,20}] (* Harvey P. Dale, Jun 04 2011 *)

PARI
```
a(n)=binomial(n^2,n*(n+1)/2)
```

a(n) = C(n^2, n*(n+1)/2) = (n^2!)/((n(n+1)/2)!*(n(n-1)/2)!).

a(n) = C(n^2, n*(n-1)/2).

More terms from Emeric Deutsch, Jul 16 2005
Offset changed to 0 (with a(0)=1), and name changed slightly by Paul D. Hanna, Jun 24 2011
Terms a(12) and beyond from Andrew Howroyd, Nov 09 2019

cp's OEIS Frontend

A135758 Catalan numbers at triangular positions: a(n) = A000108(n(n+1)/2).

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