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A051798 a(n) = (n+1)*(n+2)*(n+3)*(9n+4)/24. 5

%I

%S 1,13,55,155,350,686,1218,2010,3135,4675,6721,9373,12740,16940,22100,

%T 28356,35853,44745,55195,67375,81466,97658,116150,137150,160875,

%U 187551,217413,250705,287680,328600,373736,423368,477785,537285

%N a(n) = (n+1)*(n+2)*(n+3)*(9n+4)/24.

%C Partial sums of A007586.

%C Convolution of A000027 with A051682 (excluding 0). - _Bruno Berselli_, Dec 07 2012

%D A. 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, pps 10-20,79-94.

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

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

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

%F a(n) = C(n+3, 3)*(9*n+4)/4.

%F G.f.: (1+8*x)/(1-x)^5.

%F a(0)=1, a(1)=13, a(2)=55, a(3)=155, a(4)=350, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - _Harvey P. Dale_, Aug 19 2012

%F a(n) = A080852(9,n). - _R. J. Mathar_, Jul 28 2016

%t Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,13,55,155,350},40] (* _Harvey P. Dale_, Aug 19 2012 *)

%o (MAGMA) /* A000027 convolved with A051682 (excluding 0): */ A051682:=func<n | n*(9*n-7)/2>; [&+[(n-i+1)*A051682(i): i in [1..n]]: n in [1..35]]; // _Bruno Berselli_, Dec 07 2012

%o (PARI) a(n)=(n+1)*(n+2)*(n+3)*(9*n+4)/24 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A007586, A051682.

%Y Cf. A093644 ((9, 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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Last modified February 17 22:32 EST 2018. Contains 299297 sequences. (Running on oeis4.)