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%I #23 Feb 11 2022 04:48:57
%S 1,14,60,170,385,756,1344,2220,3465,5170,7436,10374,14105,18760,24480,
%T 31416,39729,49590,61180,74690,90321,108284,128800,152100,178425,
%U 208026,241164,278110,319145,364560,414656,469744,530145,596190
%N Partial sums of A007587.
%C 4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - _Jonathan Vos Post_, Mar 17 2006
%C Convolution of A000027 with A051624 (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 Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
%D Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
%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) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.
%F G.f.: (1+9*x)/(1-x)^5.
%F From _Amiram Eldar_, Feb 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).
%F Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)
%t Accumulate[Table[n(n+1)(10n-7)/6,{n,0,50}]] (* _Harvey P. Dale_, Nov 13 2013 *)
%o (Magma) /* A000027 convolved with A051624 (excluding 0): */ A051624:=func<n | n*(5*n-4)>; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // _Bruno Berselli_, Dec 07 2012
%Y Cf. A007587, A001622, A051624.
%Y Cf. A093645 ((10, 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
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