login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A236463 Coefficients: b(k,4,p) = Sum_{i=0..k-4} (-1)^i*C(4*p+1,i)*C(k-i,4)^p, where k = 4 + i . 3
1, 1, 16, 36, 16, 1, 1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1, 1, 608, 40136, 724320, 4961755, 15018688, 21571984, 15018688, 4961755, 724320, 40136, 608, 1, 1, 3104, 693960, 37229920, 733059110, 6501577152, 29066972368, 69830127680, 93200908410, 69830127680 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Using these coefficients we can obtain formulas for the sums Sum_{i=1..n} C(3+i,4)^p and C(n,4)^p.

Let us define:

b(k,4,p) = Sum_{i=0..k-4} C(4*p+1,i)*C(k-i,4)^p, where k=e+i .

For example:

b(4,4,p) = 1;

b(5,4,p) = 5^p - (4*p+1);

b(6,4,p) = 15^p - (4*p+1)*5^p + C(4*p+1,2);

b(7,4,p) = 35^p - (4*p+1)*15^p + C(4*p+1,2)*5^p - C(4*p+1,3);

b(8,4,p) = 70^p - (4*p+1)*35^p + C(4*p+1,2)*15^p - C(4*p+1,3)*5^p + C(4*p+1,4).

Generally if b(k,e,p) = Sum_{i=0..k-e}(-1)^i C(e*p+1,i)*C(k-i,e)^p, where k =e+i.

Sum_{i=1..n} C(e-1+i,e)^p = Sum_{i=0..e*(p-1)}b(e+i,e,p)*C(n+1+i,e*p+1), and:

C(n,e)^p = Sum_{i=0..e*(p-1)} b(e+i,e,p)*C(n+i,e*p).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..780

FORMULA

Sum_{i=1..n} C(3+i,4)^p = Sum_{i=0..4*p-4} b(4+i,4,p)*C(n+4+i,4*p+1), and

C(n,4)^p = Sum_{i=0..4*p-4} b(4+i,4,p)*C(n+i,4*p).

EXAMPLE

Sum_{i=1..n}C(3+i,4)^3 = C(n+4,13) + 112*C(n+5,13) + 1828*C(n+6,13) + 8464*C(n+7,13) + 13840*C(n+8,13) + 8464*C(n+9,13) + 1828*C(n+10,13) + 112*C(n+11,13) + C(+12,13).

C(n,4)^3 = C(n,12) + 112*C(n+1,12) + 1828*C(n+2,12) + 8464*C(n+3,12) + 13840*C(n+4,12) + 8464*C(n+5,12) + 1828*C(n+6,12) + 112*C(n+7,12) + C(n+8,12).

Coefficients triangle:

1,

1, 16, 36, 16, 1;

1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1;

1, 608, 40136, 724320, 4961755, 15018688, 21571984, 15018688, 4961755, 724320, 40136, 608, 1;

1, 3104, 693960, 37229920, 733059110, 6501577152, 29066972368, 69830127680, 93200908410, 69830127680, 29066972368, 6501577152, 733059110, 37229920, 693960, 3104, 1;

1, 15600, 11000300, 1558185200, 75073622025, 1585757994496, 16938467955200, 99825129369600, 342907451401150, 710228619472800, 903546399077256, 710228619472800, 342907451401150, 99825129369600, 16938467955200, 1585757994496, 75073622025, 1558185200, 11000300, 15600, 1;

MATHEMATICA

b[k_, 4, p_] := Sum[(-1)^i*Binomial[4*p+1, i]*Binomial[k-i, 4]^p /. k -> 4+i, {i, 0, k-4}]; row[p_] := Table[b[k, 4, p], {k, 4, 4*p}]; Table[row[p], {p, 1, 6}] // Flatten (* Jean-Fran├žois Alcover, Feb 05 2014 *)

CROSSREFS

Cf. A087127, A086023, A086024, A086025, A087107, A087108, A087109, A087110, A087111, A154283, A174266, A181544.

Sequence in context: A105509 A219316 A317818 * A070588 A250432 A183196

Adjacent sequences:  A236460 A236461 A236462 * A236464 A236465 A236466

KEYWORD

nonn,tabf

AUTHOR

Yahia Kahloune, Feb 01 2014

EXTENSIONS

a(36) corrected by Vincenzo Librandi, Feb 14 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 15 12:44 EDT 2019. Contains 327078 sequences. (Running on oeis4.)