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 A154322 #26 Jan 19 2023 19:15:51
%S A154322 1,2,4,10,26,62,133,260,471,802,1298,2014,3016,4382,6203,8584,11645,
%T A154322 15522,20368,26354,33670,42526,53153,65804,80755,98306,118782,142534,
%U A154322 169940,201406,237367,278288,324665,377026,435932,501978,575794,658046,749437,850708
%N A154322 a(n) = 1 + n + binomial(n+3,5).
%C A154322 Row sums of number triangle A113582.
%C A154322 It appears that the sequence is the pairwise sum of terms in A101338 and A000389 with offsets as follows:
%C A154322 1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, ... =
%C A154322 1, 2, 4, 9, 20, 41, 77, 134, 219, 340, 506, ... +
%C A154322 0, 0, 0, 1, 6, 21, 56, 126, 252, 462, 792, ...
%C A154322 - _Gary W. Adamson_, Oct 08 2015
%H A154322 G. C. Greubel, <a href="/A154322/b154322.txt">Table of n, a(n) for n = 0..1000</a>
%H A154322 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A154322 G.f.: (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1-x)^6;
%F A154322 a(n) = n + 1 + Sum_{k=0..n} binomial(k+1,2) * binomial(n-k+1,2).
%F A154322 a(n) = (n+1)*(n^4 +4*n^3 +n^2 -6*n +120)/120.
%F A154322 a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6) for n>5. - _Wesley Ivan Hurt_, Oct 08 2015
%F A154322 E.g.f.: (1/120)*(120 + 120*x + 60*x^2 + 60*x^3 + 15*x^4 + x^5)*exp(x). - _G. C. Greubel_, Sep 10 2016
%p A154322 A154322:=n->1+n+binomial(n+3,5): seq(A154322(n), n=0..50); # _Wesley Ivan Hurt_, Oct 08 2015
%t A154322 CoefficientList[Series[(1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1 - x)^6, {x, 0, 40}], x] (* _Wesley Ivan Hurt_, Oct 08 2015 *)
%t A154322 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 2, 4, 10, 26, 62}, 50] (* _Vincenzo Librandi_, Oct 09 2015 *)
%t A154322 Table[ (n + 1)*(n^4 + 4*n^3 + n^2 - 6*n + 120)/120 , {n, 0, 25}] (* _G. C. Greubel_, Sep 10 2016 *)
%t A154322 Table[1+n+Binomial[n+3,5],{n,0,40}] (* _Harvey P. Dale_, Jan 19 2023 *)
%o A154322 (Magma) [1+n+Binomial(n+3,5) : n in [0..50]]; // _Wesley Ivan Hurt_, Oct 08 2015
%o A154322 (Magma) I:=[1,2,4,10,26,62]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Oct 09 2015
%o A154322 (PARI) Vec((1-4*x+7*x^2-4*x^3+x^4)/(1-x)^6 + O(x^100)) \\ _Altug Alkan_, Oct 18 2015
%Y A154322 Cf. A000389, A101338, A113582.
%K A154322 easy,nonn
%O A154322 0,2
%A A154322 _Paul Barry_, Jan 07 2009