A090162 - OEIS

%I #45 Jan 05 2025 19:51:37

%S 1,3003,61218182743304701891431482520

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

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

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

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

%H Hugo Pfoertner, <a href="/A090162/b090162.txt">Table of n, a(n) for n = 1..5</a>

%H A. I. Shirshov, <a href="https://bookstore.ams.org/view?ProductCode=MAWRLD/14">On the equation C(n, m) = C(n+1, m-1)</a>, chapter 10 in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am. Math. Soc., 1999, pp. 83-86

%H D. Singmaster, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/singmaster.pdf">Repeated binomial coefficients and Fibonacci numbers</a>, Fibonacci Quarterly, 13 (1975), 295-298.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PascalsTriangle.html">Pascal's Triangle</a>

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

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

%F 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

%p a := proc(n) local a,b,s,p; s:= 1+sqrt(5); p:=16^n;

%p a := 4-2*p*s^(-4*n-1)+(s+2)*s^(4*n-1)/p:

%p b := 1+p*((s-2)^(1-4*n)/2-s^(-1-4*n)*(2+s)):

%p GAMMA(a/5)/(GAMMA(b/5)*GAMMA(1+(a-b)/5)) end:

%p digits := [1, 4, 29, 205, 1412]: A := n -> round(evalf(a(n),digits[n]+10)):

%p A(4); # _Peter Luschny_, Jul 15 2017

%t Table[Binomial[Fibonacci[2k]Fibonacci[2k+1],Fibonacci[2k-1] Fibonacci[2k]-1], {k,4}] (* _Harvey P. Dale_, Aug 18 2011 *)

%o (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

%o (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

%Y Subsequence of A003015.

%Y Cf. A081016, A089508, A062527, A206351.

%K nonn,nice,changed

%O 1,2

%A _Eric W. Weisstein_, Nov 23 2003 and _Wolfdieter Lang_, Dec 01 2003