This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A191495 #26 Jun 30 2025 15:29:50
%S A191495 1,128,3281,32768,195313,839808,2882401,8388608,21523361,50000000,
%T A191495 107179441,214990848,407865361,737894528,1281445313,2147483648,
%U A191495 3487878721,5509980288,8491781521,12800000000,18911429681,27437936768,39155492641,55037657088,76293945313,104413532288
%N A191495 Number of compositions of even natural numbers into 8 parts <= n.
%C A191495 Number of ways of placing an even number of indistinguishable objects in 8 distinguishable boxes with the condition that in each box can be at most n objects.
%H A191495 Vincenzo Librandi, <a href="/A191495/b191495.txt">Table of n, a(n) for n = 0..1000</a>
%H A191495 Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%H A191495 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (8,-27,48,-42,0,42,-48,27,-8,1).
%F A191495 a(n) = ( (n+1)^8 + (1 + (-1)^n)/2 )/2.
%F A191495 G.f.: ( -1 - 120*x - 2284*x^2 - 9928*x^3 - 15654*x^4 - 9928*x^5 - 2284*x^6 - 120*x^7 - x^8 ) / ( (1+x)*(x-1)^9 ). - _R. J. Mathar_, Jun 06 2011
%F A191495 a(n) = 8*a(n-1) - 27*a(n-2) + 48*a(n-3) - 42*a(n-4) + 42*a(n-6) - 48*a(n-7) + 27*a(n-8) - 8*a(n-9) + a(n-10); a(0)=1, a(1)=128, a(2)=3281, a(3)=32768, a(4)=195313, a(5)=839808, a(6)=2882401, a(7)=8388608, a(8)=21523361, a(9)=50000000. - _Harvey P. Dale_, Mar 13 2013
%e A191495 a(1)=128 compositions of even natural numbers into 8 parts no greater than 1 are
%e A191495 (0,0,0,0,0,0,0,0) --> 8!/(8!0!) = 1
%e A191495 (0,0,0,0,0,0,1,1) --> 8!/(6!2!) = 28
%e A191495 (0,0,0,0,1,1,1,1) --> 8!/(4!4!) = 70
%e A191495 (0,0,1,1,1,1,1,1) --> 8!/(2!6!) = 28
%e A191495 (1,1,1,1,1,1,1,1) --> 8!/(0!8!) = 1
%t A191495 Table[1/2*((n + 1)^8 + (1 + (-1)^n)*1/2), {n, 0, 25}]
%t A191495 LinearRecurrence[{8,-27,48,-42,0,42,-48,27,-8,1},{1,128,3281,32768,195313,839808,2882401,8388608,21523361,50000000},30] (* _Harvey P. Dale_, Mar 13 2013 *)
%o A191495 (Magma) [ 1/2*((n + 1)^8 + (1 + (-1)^n)*1/2): n in [0..35]]; // _Vincenzo Librandi_, Jun 06 2011
%o A191495 (PARI) a(n)=((n+1)^8+1)>>1 \\ _Charles R Greathouse IV_, Jun 06 2011
%Y A191495 Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191489 (6 parts), A191494 (7 parts).
%K A191495 nonn,easy
%O A191495 0,2
%A A191495 _Adi Dani_, Jun 03 2011