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 A051740 #45 Jun 01 2025 18:05:54
%S A051740 1,11,45,125,280,546,966,1590,2475,3685,5291,7371,10010,13300,17340,
%T A051740 22236,28101,35055,43225,52745,63756,76406,90850,107250,125775,146601,
%U A051740 169911,195895,224750,256680,291896,330616,373065,419475,470085,525141
%N A051740 Partial sums of A007584.
%C A051740 Convolution of A000027 with A001106 (excluding 0). - _Bruno Berselli_, Dec 07 2012
%D A051740 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D A051740 Murray R.Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
%H A051740 G. C. Greubel, <a href="/A051740/b051740.txt">Table of n, a(n) for n = 0..1000</a>
%H A051740 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A051740 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A051740 a(n) = binomial(n+3, 3)*(7*n+4)/4.
%F A051740 a(n) = (7*n+4)*binomial(n+3, 3)/4.
%F A051740 G.f.: (1+6*x)/(1-x)^5.
%F A051740 a(n) = A080852(7,n). - _R. J. Mathar_, Jul 28 2016
%F A051740 E.g.f.: (4! + 240*x + 288*x^2 + 88*x^3 + 7*x^4)*exp(x)/4!. - _G. C. Greubel_, Aug 29 2019
%p A051740 seq((7*n+4)*binomial(n+3,3)/4, n=0..40); # _G. C. Greubel_, Aug 29 2019
%t A051740 Table[(7*n+4)*Binomial[n+3,3]/4, {n,0,40}] (* _G. C. Greubel_, Aug 29 2019 *)
%t A051740 LinearRecurrence[{5,-10,10,-5,1},{1,11,45,125,280},40] (* _Harvey P. Dale_, May 18 2023 *)
%o A051740 (Magma) /* A000027 convolved with A001106 (excluding 0): */ A001106:=func<n | n*(7*n-5)/2>; [&+[(n-i+1)*A001106(i): i in [1..n]]: n in [1..36]]; // _Bruno Berselli_, Dec 07 2012
%o A051740 (PARI) vector(40, n, (7*n-3)*binomial(n+2,3)/4) \\ _G. C. Greubel_, Aug 29 2019
%o A051740 (Sage) [(7*n+4)*binomial(n+3,3)/4 for n in (0..40)] # _G. C. Greubel_, Aug 29 2019
%o A051740 (GAP) List([0..40], n-> (7*n+4)*Binomial(n+3,3)/4); # _G. C. Greubel_, Aug 29 2019
%Y A051740 Cf. A001106, A007584.
%Y A051740 Cf. A093564 ((7, 1) Pascal, column m=4).
%Y A051740 Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
%K A051740 nonn,easy
%O A051740 0,2
%A A051740 _Barry E. Williams_, Dec 07 1999