A172066 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=9.

1, 10, 67, 376, 1912, 9142, 41941, 186880, 815083, 3498146, 14827487, 62236064, 259187048, 1072567256, 4415408372, 18098359424, 73915594466, 300958990724, 1222228100590, 4952609171080, 20030298812596, 80876902778482

Offset

0,2

Comments

This sequence is the 9th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The Maple programs give the first diagonals of this array.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{j=0..n} (-1)^j * binomial(2*n+k-j,n-j), with k=9.

a(n) ~ 2^(2*n+10)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014

Conjecture: 2*n*(n+9)*(n+5)*a(n) -(7*n^3+94*n^2+427*n+672)*a(n-1) -2*(2*n+7)*(n+6)*(n+4)*a(n-2)=0. - R. J. Mathar, Feb 19 2016

Example

a(4) = C(17,4) - C(16,3) + C(15,2) - C(14,1) + C(13,0) = 17*4*5*7 - 16*5*7 + 105 - 14 + 1 = 5*7*(68-16) + 92 = 1912.

Maple

for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od;
# 2nd program
for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;

Mathematica

CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *)

Prog

(PARI) k=9; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 17 2019
(Magma) k:=9; m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 17 2019
(Sage) k=9; m=30; a=((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k ).series(x, m+2).coefficients(x, sparse=False); a[0:m] # G. C. Greubel, Feb 17 2019

Crossrefs

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172067 (k=10).

Sequence in context: A266443 A108275 A086443 * A026844 A026875 A026868

Adjacent sequences: A172063 A172064 A172065 * A172067 A172068 A172069

Keyword

easy,nonn

Author

Richard Choulet, Jan 24 2010

Status

approved