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

1, 13, 55, 155, 350, 686, 1218, 2010, 3135, 4675, 6721, 9373, 12740, 16940, 22100, 28356, 35853, 44745, 55195, 67375, 81466, 97658, 116150, 137150, 160875, 187551, 217413, 250705, 287680, 328600, 373736, 423368, 477785, 537285

Offset: 0

Author

Barry E. Williams, Dec 11 1999

Keywords

• nonn
• easy

Comments

Partial sums of A007586.

Convolution of A000027 with A051682 (excluding 0). - Bruno Berselli, Dec 07 2012

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
• Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

Links

• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A007586, A051682.

Cf. A093644 ((9, 1) Pascal, column m=4).

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

Programs

Magma

/* A000027 convolved with A051682 (excluding 0): */ A051682:=func; [&+[(n-i+1)*A051682(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Mathematica

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 *)

PARI

a(n)=(n+1)*(n+2)*(n+3)*(9*n+4)/24 \\ Charles R Greathouse IV, Oct 07 2015

Formula

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

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

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

a(n) = A080852(9,n). - R. J. Mathar, Jul 28 2016