A051799 Partial sums of A007587.

1, 14, 60, 170, 385, 756, 1344, 2220, 3465, 5170, 7436, 10374, 14105, 18760, 24480, 31416, 39729, 49590, 61180, 74690, 90321, 108284, 128800, 152100, 178425, 208026, 241164, 278110, 319145, 364560, 414656, 469744, 530145, 596190

Offset: 0

Author

Barry E. Williams, Dec 11 1999

Keywords

• nonn
• easy

Comments

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

Convolution of A000027 with A051624 (excluding 0). - Bruno Berselli, Dec 07 2012

References

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

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

Crossrefs

Cf. A007587, A001622, A051624.

Cf. A093645 ((10, 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 A051624 (excluding 0): */ A051624:=func; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Mathematica

Accumulate[Table[n(n+1)(10n-7)/6,{n,0,50}]] (* Harvey P. Dale, Nov 13 2013 *)

Formula

a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.

G.f.: (1+9*x)/(1-x)^5.

From Amiram Eldar, Feb 11 2022: (Start)

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).

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)