A051877 Partial sums of A051740.

1, 12, 57, 182, 462, 1008, 1974, 3564, 6039, 9724, 15015, 22386, 32396, 45696, 63036, 85272, 113373, 148428, 191653, 244398, 308154, 384560, 475410, 582660, 708435, 855036, 1024947, 1220842, 1445592, 1702272, 1994168, 2324784, 2697849, 3117324, 3587409

Offset

0,2

Comments

Convolution of triangular numbers (A000217) and enneagonal numbers (A001106). - Bruno Berselli, Jul 21 2015

References

A. 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. 1-16.

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 (6,-15,20,-15,6,-1).

Formula

a(n) = C(n+4, 4)*(7*n+5)/5.

G.f.: (1+6*x)/(1-x)^6.

From Wesley Ivan Hurt, May 02 2015: (Start)

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6).

a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(7*n+5)/120. (End)

E.g.f.: (5! +1320*x +2040*x^2 +920*x^3 +145*x^4 +7*x^5)*exp(x)/5!

Maple

A051877:=n->binomial(n+4, 4)*(7*n+5)/5: seq(A051877(n), n=0..40); # Wesley Ivan Hurt, May 02 2015

Mathematica

Table[(n+1)(n+2)(n+3)(n+4)(7n+5)/120, {n, 0, 40}] (* Vincenzo Librandi, May 03 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 12, 57, 182, 462, 1008}, 40] (* Harvey P. Dale, May 05 2022 *)

Prog

(Magma) [(n+1)*(n+2)*(n+3)*(n+4)*(7*n+5)/120 : n in [0..40]]; // Wesley Ivan Hurt, May 02 2015
(PARI) vector(40, n, (7*n-2)*binomial(n+3, 4)/5) \\ G. C. Greubel, Aug 29 2019
(Sage) [(7*n+5)*binomial(n+4, 4)/5 for n in (0..40)] # G. C. Greubel, Aug 29 2019
(GAP) List([0..40], n-> (7*n+5)*Binomial(n+4, 4)/5); # G. C. Greubel, Aug 29 2019

Crossrefs

Cf. A051740; A000217, A001106.

Cf. A093564 ((7, 1) Pascal, column m=5).

Sequence in context: A212682 A212134 A071270 * A212065 A121693 A190297

Adjacent sequences: A051874 A051875 A051876 * A051878 A051879 A051880

Keyword

nonn,easy

Author

Barry E. Williams, Dec 14 1999

Status

approved