A090162 Values of binomial(Fibonacci(2k)*Fibonacci(2k+1),Fibonacci(2k-1)*Fibonacci(2k)-1).

1, 3003, 61218182743304701891431482520

Offset: 1

Author

Eric W. Weisstein, Nov 23 2003 and Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• nice

Comments

These numbers are known to occur at least six times in Pascal's triangle.

The next term is approximately 3.537 * 10^204 and is in the b-file.

The numbers of digits in a(n), n >= 1, are given in A100022.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..5
• A. I. Shirshov, On the equation C(n, m) = C(n+1, m-1), chapter 10 in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am. Math. Soc., 1999, pp. 83-86
• D. Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly, 13 (1975), 295-298.
• Eric Weisstein's World of Mathematics, Pascal's Triangle

Crossrefs

Subsequence of A003015.

Cf. A081016, A089508, A062527, A206351.

Programs

Maple

a := proc(n) local a,b,s,p; s:= 1+sqrt(5); p:=16^n;
a := 4-2*p*s^(-4*n-1)+(s+2)*s^(4*n-1)/p:
b := 1+p*((s-2)^(1-4*n)/2-s^(-1-4*n)*(2+s)):
GAMMA(a/5)/(GAMMA(b/5)*GAMMA(1+(a-b)/5)) end:
digits := [1, 4, 29, 205, 1412]: A := n -> round(evalf(a(n),digits[n]+10)):
A(4); # Peter Luschny, Jul 15 2017

Mathematica

Table[Binomial[Fibonacci[2k]Fibonacci[2k+1],Fibonacci[2k-1] Fibonacci[2k]-1], {k,4}] (* Harvey P. Dale, Aug 18 2011 *)

PARI

A090162(n)=binomial(fibonacci(2*n+1)*fibonacci(2*n),fibonacci(2*n-1)*fibonacci(2*n)-1) \\ M. F. Hasler, Feb 17 2023

Python

def A090162(n): return C(A000045(2*n+1)*A000045(2*n),A000045(2*n-1)*A000045(2*n)-1) # See A007318 for C(.,.). - M. F. Hasler, Feb 17 2023

Formula

a(n) = binomial(A089508(n), A081016(n-1)).

a(n) = binomial(A089508(n)+1, A081016(n-1)-1).

a(n) = Gamma(x)/(Gamma(y)*Gamma(1+x-y)) with x = A206351(n+1) and y = A081016(n-1). - Peter Luschny, Jul 15 2017