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, 602175, 672771, 749398, 832390, 922090

Offset

0,2

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.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 211.

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

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

Links

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000

Index to sequences related to pyramidal numbers

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

E.g.f.: exp(x)*(24 + 288*x + 360*x^2 + 112*x^3 + 9*x^4)/24. - Stefano Spezia, Oct 29 2025

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 *)

Prog

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

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.

Sequence in context: A158485 A274973 A005902 * A206372 A290396 A061161

Adjacent sequences: A051795 A051796 A051797 * A051799 A051800 A051801

Keyword

nonn,easy

Author

Barry E. Williams, Dec 11 1999

Extensions

a(34)-a(38) from Stefano Spezia, Oct 29 2025

Status

approved