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A051797 Partial sums of A007585. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index to sequences related to pyramidal numbers

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

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

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

(Sage) [(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. A001107, A007585.

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 * A333276 A145886 A268351

Adjacent sequences:  A051794 A051795 A051796 * A051798 A051799 A051800

KEYWORD

nonn,easy

AUTHOR

Barry E. Williams, Dec 11 1999

STATUS

approved

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Last modified February 27 04:40 EST 2021. Contains 341649 sequences. (Running on oeis4.)