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

Author

Barry E. Williams, Dec 14 1999

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A051740; A000217, A001106.

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

Programs

GAP

List([0..40], n-> (7*n+5)*Binomial(n+4,4)/5); # G. C. Greubel, Aug 29 2019

Magma

[(n+1)*(n+2)*(n+3)*(n+4)*(7*n+5)/120 : n in [0..40]]; // Wesley Ivan Hurt, May 02 2015

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 *)

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

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!