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Partial sums of A007585.
8

%I #45 Feb 11 2022 04:48:46

%S 1,12,50,140,315,616,1092,1800,2805,4180,6006,8372,11375,15120,19720,

%T 25296,31977,39900,49210,60060,72611,87032,103500,122200,143325,

%U 167076,193662,223300,256215,292640,332816,376992,425425,478380,536130

%N Partial sums of A007585.

%C a(n-1) is the n-th antidiagonal sum of the convolution array A213835. - _Clark Kimberling_, Jul 04 2012

%C Convolution of A000027 with A001107 (excluding 0). - _Bruno Berselli_, Dec 07 2012

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%D Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

%D Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

%H G. C. Greubel, <a href="/A051797/b051797.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.

%F a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6.

%F G.f.: (1+7*x)/(1-x)^5.

%F a(n) = A080851(8,n). - _R. J. Mathar_, Jul 28 2016

%F 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 From _Amiram Eldar_, Feb 11 2022: (Start)

%F Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10.

%F Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End)

%p seq((2*n+1)*binomial(n+3,3), n=0..40); # _G. C. Greubel_, Aug 30 2019

%t 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 (Magma) /* A000027 convolved with A001107 (excluding 0): */

%o 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 (Magma) [(2*n+1)*Binomial(n+3,3): n in [0..40]]; // _G. C. Greubel_, Aug 30 2019

%o (PARI) vector(40, n, (2*n-1)*binomial(n+2,3)) \\ _G. C. Greubel_, Aug 30 2019

%o (Sage) [(2*n+1)*binomial(n+3,3) for n in (0..40)] # _G. C. Greubel_, Aug 30 2019

%o (GAP) List([0..40], n-> (2*n+1)*Binomial(n+3,3)) # _G. C. Greubel_, Aug 30 2019

%Y Cf. A000027, A001107, A007585, A080851.

%Y Cf. A093565 ((8, 1) Pascal, column m=4).

%Y Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

%K nonn,easy

%O 0,2

%A _Barry E. Williams_, Dec 11 1999