A054487 a(n) = (3*n+4)*binomial(n+7, 7)/4.

1, 14, 90, 390, 1320, 3762, 9438, 21450, 45045, 88660, 165308, 294372, 503880, 833340, 1337220, 2089164, 3187041, 4758930, 6970150, 10031450, 14208480, 19832670, 27313650, 37153350, 49961925, 66475656, 87576984, 114316840

Offset: 0

Author

Barry E. Williams, May 06 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
• E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A034265.

Cf. A093563 ((6, 1) Pascal, column m=8).

Programs

GAP

List([0..40], n-> (3*n+4)*Binomial(n+7, 7)/4 ); # G. C. Greubel, Jan 19 2020

Magma

[((3*n+4)*Binomial(n+7,7))/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

Maple

seq( (3*n+4)*binomial(n+7,7)/4, n=0..40); # G. C. Greubel, Jan 19 2020

Mathematica

CoefficientList[Series[(1+5x)/(1-x)^9, {x,0,40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Table[6*Binomial[n+8,8] -5*Binomial[n+7,7], {n,0,40}] (* G. C. Greubel, Jan 19 2020 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,14,90,390,1320,3762,9438,21450,45045},30] (* Harvey P. Dale, Jul 19 2022 *)

PARI

a(n) = (3*n+4)*binomial(n+7, 7)/4; \\ Michel Marcus, Sep 07 2017

Sage

[(3*n+4)*binomial(n+7, 7)/4 for n in (0..40)] # G. C. Greubel, Jan 19 2020

Formula

G.f.: (1+5*x)/(1-x)^9.

From G. C. Greubel, Jan 19 2020: (Start)

a(n) = 6*binomial(n+8, 8) - 5*binomial(n+7, 7).

E.g.f.: (20160 +262080*x +635040*x^2 +540960*x^3 +205800*x^4 +38808*x^5 +3724*x^6 +172*x^7 +3*x^8)*exp(x)/20160. (End)

a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Wesley Ivan Hurt, Jun 07 2021

Extensions

Corrected and extended by James Sellers, May 10 2000