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A237202 Coefficients : b(k,5,p) = Sum_{i=0..(k-5)}(-1)^i*C(5*p+1,i)*C(k-i,5)^p ; where k = 5+i. 3
1, 1, 25, 100, 100, 25, 1, 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1, 1, 1275, 167475, 6021225, 84646275, 554083761, 1858142825, 3363309675, 3363309675, 1858142825, 554083761, 84646275 (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(4+i,5)^p and C(n,5)^p. Let us define

b(k,5,p) = Sum_{i=0..k-5}C(5*p+1,i)* C(k-i,5)^p ;where k=5+i .

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+e+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

G. C. Greubel, Table of n, a(n) for the first 25 rows, flattened

FORMULA

Then we have formulas :

Sum_{i=1..n} C(4+i,5)^p = Sum_{i=0..5*(p-1)} b(5+i,5,p)*C(n+5+i,5*p+1) and

C(n,5)^p = Sum_{i=0..5*(p-1)} b(5+i,5,p)*C(n+i,5*p).

EXAMPLE

For example:

b(5,5,p) = 1 ;

b(6,5,p) = 6^p - (5*p+1) ;

b(7,5,p) = 21^p - (5*p+1)*6^p + C(5*p+1,2) ;

b(8,5,p) = 56^p - (5*p+1)*21^p + C(5*p+1,2)*6^p - C(5*p+1,3) ;

b(9,5,p) = 126^p - (5*p+1)*56^p + C(5*p+1,2)*21^p - C(5*p+1,3)*6^p  + C(5*p+1,4) .

Coefficients triangle :

1;

1, 25, 100, 100, 25, 1;

1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1;

1, 1275, 167475, 6021225, 84646275, 554083761, 1858142825, 3363309675, 3363309675, 1858142825, 554083761, 84646275, 6021225, 167475, 125, 1;

1, 7750, 3882250, 447069750, 18746073375, 359033166276, 3575306548500, 20052364456500, 66640122159000, 135424590593500, 171219515211316, 135424590593500, 66640122159000, 20052364456500, 3575306548500, 359033166276, 18746073375, 447069750, 3882250, 7750, 1;

example:

Sum_{i=1..n}C(4+i,5)^p = C(n+5,16) +200*C(n+6,16) +5925*(n+7,16) + 52800*C(n+8,16) + 182700*C(n+9,16) + 273504*C(n+10,16) + 182700*C(n+11,16) + 52800*C(n+12,16) + 5925*C(n+13,16) + 200*C(n+14,16) + C(n+15,16) ;

C(n,5)^p = C(n,15) + 200*C(n+1,15) + 5925*C(n+2,15) + 52800*C(n+3,15) + 182700*C(n+4,15) + 273504*C(n+5,15) + 182700*C(n+6,15) + 52800*C(n+7,15) + 5925*C(n+8,15) + 200*C(n+9,15)  C(n+10,15) .

MATHEMATICA

b[k_, 5, p_] := Sum[(-1)^i*Binomial[5*p+1, i]*Binomial[k-i, 5]^p /. k -> 5+i, {i, 0, k-5}]; row[p_] := Table[b[k, 5, p], {k, 5, 5*p}]; Table[row[p], {p, 1, 5}] // Flatten (* Jean-Fran├žois Alcover, Feb 05 2014 *)

CROSSREFS

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

A236463.

Sequence in context: A266818 A158547 A144854 * A198385 A134422 A016850

Adjacent sequences:  A237199 A237200 A237201 * A237203 A237204 A237205

KEYWORD

nonn,tabf

AUTHOR

Yahia Kahloune, Feb 05 2014

STATUS

approved

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Last modified February 17 21:35 EST 2020. Contains 332006 sequences. (Running on oeis4.)