A051797 Partial sums of A007585.

1, 12, 50, 140, 315, 616, 1092, 1800, 2805, 4180, 6006, 8372, 11375, 15120, 19720, 25296, 31977, 39900, 49210, 60060, 72611, 87032, 103500, 122200, 143325, 167076, 193662, 223300, 256215, 292640, 332816, 376992, 425425, 478380, 536130

Offset: 0

Author

Barry E. Williams, Dec 11 1999

Keywords

• nonn
• easy

Comments

a(n-1) is the n-th antidiagonal sum of the convolution array A213835. - Clark Kimberling, Jul 04 2012

Convolution of A000027 with A001107 (excluding 0). - Bruno Berselli, Dec 07 2012

References

• Albert 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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
• Index to sequences related to pyramidal numbers.

Crossrefs

Cf. A000027, A001107, A007585, A080851.

Cf. A093565 ((8, 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

GAP

List([0..40], n-> (2*n+1)*Binomial(n+3,3)); # G. C. Greubel, Aug 30 2019

Magma

/* A000027 convolved with A001107 (excluding 0): */
A001107:=func; [&+[(n-i+1)*A001107(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Magma

[(2*n+1)*Binomial(n+3,3): n in [0..40]]; // G. C. Greubel, Aug 30 2019

Maple

seq((2*n+1)*binomial(n+3,3), n=0..40); # G. C. Greubel, Aug 30 2019

Mathematica

Table[(2*n+1)*Binomial[n+3,3], {n,0,40}] (*  Vladimir Joseph Stephan Orlovsky, Apr 19 2011, modified by G. C. Greubel, Aug 30 2019 *)

PARI

vector(40, n, (2*n-1)*binomial(n+2,3)) \\ G. C. Greubel, Aug 30 2019

Sage

[(2*n+1)*binomial(n+3,3) for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6.

G.f.: (1+7*x)/(1-x)^5.

a(n) = A080851(8,n). - R. J. Mathar, Jul 28 2016

E.g.f.: (6 + 66*x + 81*x^2 + 25*x^3 + 2*x^4)*exp(x)/6. - G. C. Greubel, Aug 30 2019

From Amiram Eldar, Feb 11 2022: (Start)

Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10.

Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End)