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, 598956, 667147, 741000, 820820

Offset

0,2

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.

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

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.

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)

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 *)

Prog

(Magma) /* A000027 convolved with A001107 (excluding 0): */
A001107:=func<n | n*(4*n-3)>; [&+[(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
(PARI) vector(40, n, (2*n-1)*binomial(n+2, 3)) \\ G. C. Greubel, Aug 30 2019
(SageMath) [(2*n+1)*binomial(n+3, 3) for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> (2*n+1)*Binomial(n+3, 3)); # G. C. Greubel, Aug 30 2019

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.

Sequence in context: A063491 A248230 A083559 * A342482 A333276 A145886

Adjacent sequences: A051794 A051795 A051796 * A051798 A051799 A051800

Keyword

nonn,easy

Author

Barry E. Williams, Dec 11 1999

Status

approved