A055843 Expansion of (1+3*x)/(1-x)^10.

1, 13, 85, 385, 1375, 4147, 11011, 26455, 58630, 121550, 238238, 445094, 797810, 1379210, 2309450, 3759074, 5965487, 9253475, 14060475, 20967375, 30735705, 44352165, 63081525, 88529025, 122713500, 168152556, 227961228, 305965660, 406833460, 536222500, 700950052

Offset: 0

Author

Barry E. Williams, May 30 2000

Keywords

• easy
• nonn

Comments

Partial sums of A052181.

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A052181.

Cf. A093561 ((4, 1) Pascal, column m=9).

Programs

GAP

List([0..30], n-> (4*n+9)*Binomial(n+8, 8)/9 ); # G. C. Greubel, Jan 21 2020

Magma

[(4*n+9)*Binomial(n+8, 8)/9: n in [0..30]]; // G. C. Greubel, Jan 21 2020

Maple

seq( (4*n+9)*binomial(n+8, 8)/9, n=0..30); # G. C. Greubel, Jan 21 2020

Mathematica

Table[4*Binomial[n+9,9] - 3*Binomial[n+8,8], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)

PARI

vector(31, n, (4*n+5)*binomial(n+7, 8)/9) \\ G. C. Greubel, Jan 21 2020

Sage

[(4*n+9)*binomial(n+8, 8)/9 for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = (4*n+9)*binomial(n+8, 8)/9.

G.f.: (1+3*x)/(1-x)^10.

a(n) = 4*binomial(n+9,9) - 3*binomial(n+8,8). - G. C. Greubel, Jan 21 2020

Sum_{n>=0} 1/a(n) = 9437184*Pi/24035 + 56623104*log(2)/24035 - 482087736/168245. - Amiram Eldar, Feb 17 2023