A105938 a(n) = binomial(n+2,2)*binomial(n+5,2).
10, 45, 126, 280, 540, 945, 1540, 2376, 3510, 5005, 6930, 9360, 12376, 16065, 20520, 25840, 32130, 39501, 48070, 57960, 69300, 82225, 96876, 113400, 131950, 152685, 175770, 201376, 229680, 260865, 295120, 332640, 373626, 418285, 466830, 519480, 576460
Offset: 0
Examples
If n=0 then C(0+2,0)*C(0+5,2) = C(2,0)*C(5,2) = 1*10 = 10. If n=9 then C(9+2,9)*C(9+5,2) = C(11,9)*C(14,2) = 55*91 = 5005.
Links
- Nathaniel Johnston, Table of n, a(n) for n = 0..5000
- Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Programs
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Magma
A105938:= func< n | 30*Binomial(n+5,5)/(n+3) >; [A105938(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025
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Maple
a:= n-> binomial(n+2,n)*binomial(n+5,2): seq(a(n), n=0..40); # Alois P. Heinz, Oct 16 2008
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Mathematica
Table[n(n+1)(n+3)(n+4)/4, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *) Table[Binomial[n + 2, n] Binomial[n + 5, 2], {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {10, 45, 126, 280, 540}, 40] (* Harvey P. Dale, Sep 05 2013 *) -
SageMath
def A105938(n): return 30*binomial(n+5,5)//(n+3) print([A105938(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025
Formula
G.f.: (10 - 5*x + x^2)/(1-x)^5. - Alois P. Heinz, Oct 16 2008
a(0)=10, a(1)=45, a(2)=126, a(3)=280, a(4)=540; for n>4, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 05 2013
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 5/36.
Sum_{n>=0} (-1)^n/a(n) = 1/12. (End)
From G. C. Greubel, Mar 11 2025: (Start)
a(n) = 30*A000389(n+5)/(n+3).
E.g.f.: (1/4)*(40 + 140*x + 92*x^2 + 18*x^3 + x^4)*exp(x). (End)