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 A051797 #48 May 30 2025 11:02:16
%S A051797 1,12,50,140,315,616,1092,1800,2805,4180,6006,8372,11375,15120,19720,
%T A051797 25296,31977,39900,49210,60060,72611,87032,103500,122200,143325,
%U A051797 167076,193662,223300,256215,292640,332816,376992,425425,478380,536130
%N A051797 Partial sums of A007585.
%C A051797 a(n-1) is the n-th antidiagonal sum of the convolution array A213835. - _Clark Kimberling_, Jul 04 2012
%C A051797 Convolution of A000027 with A001107 (excluding 0). - _Bruno Berselli_, Dec 07 2012
%D A051797 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D A051797 Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
%D A051797 Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
%H A051797 G. C. Greubel, <a href="/A051797/b051797.txt">Table of n, a(n) for n = 0..1000</a>
%H A051797 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%H A051797 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%F A051797 a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6.
%F A051797 G.f.: (1+7*x)/(1-x)^5.
%F A051797 a(n) = A080851(8,n). - _R. J. Mathar_, Jul 28 2016
%F A051797 E.g.f.: (6 + 66*x + 81*x^2 + 25*x^3 + 2*x^4)*exp(x)/6. - _G. C. Greubel_, Aug 30 2019
%F A051797 From _Amiram Eldar_, Feb 11 2022: (Start)
%F A051797 Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10.
%F A051797 Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End)
%p A051797 seq((2*n+1)*binomial(n+3,3), n=0..40); # _G. C. Greubel_, Aug 30 2019
%t A051797 Table[(2*n+1)*Binomial[n+3,3], {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 19 2011, modified by _G. C. Greubel_, Aug 30 2019 *)
%o A051797 (Magma) /* A000027 convolved with A001107 (excluding 0): */
%o A051797 A001107:=func<n | n*(4*n-3)>; [&+[(n-i+1)*A001107(i): i in [1..n]]: n in [1..35]]; // _Bruno Berselli_, Dec 07 2012
%o A051797 (Magma) [(2*n+1)*Binomial(n+3,3): n in [0..40]]; // _G. C. Greubel_, Aug 30 2019
%o A051797 (PARI) vector(40, n, (2*n-1)*binomial(n+2,3)) \\ _G. C. Greubel_, Aug 30 2019
%o A051797 (Sage) [(2*n+1)*binomial(n+3,3) for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o A051797 (GAP) List([0..40], n-> (2*n+1)*Binomial(n+3,3)); # _G. C. Greubel_, Aug 30 2019
%Y A051797 Cf. A000027, A001107, A007585, A080851.
%Y A051797 Cf. A093565 ((8, 1) Pascal, column m=4).
%Y A051797 Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
%K A051797 nonn,easy
%O A051797 0,2
%A A051797 _Barry E. Williams_, Dec 11 1999