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 A237252 Irregular triangle read by rows, giving coefficients b(k,6,p) = Sum_{i=0..k-6}(-1)^i*C(6*p+1,i)*C(k-i,6), where k = 6+i . 2

%I

%S 1,1,36,225,400,225,36,1,1,324,15606,233300,1424925,4050864,5703096,

%T 4050864,1424925,233300,15606,324,1,1,2376,554931,35138736,879018750,

%U 10490842656,66555527346,239677178256,509723668476

%N Irregular triangle read by rows, giving coefficients b(k,6,p) = Sum_{i=0..k-6}(-1)^i*C(6*p+1,i)*C(k-i,6), where k = 6+i .

%C Using these coefficients we can obtain formulas for the sums Sum_{i=1..n}C(5+i,6)^p and C(n,6)^p.

%C Let us define b(k,6,p) = Sum_{i=0..k-6}C(6*p+1,i)*C(k-i,6)^p ; where k =6+i .

%C 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.

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

%H G. C. Greubel, <a href="/A237252/b237252.txt">Table of n, a(n) for the first 25 rows, flattened</a>

%F Then we have formulas :

%F Sum_{i=1..n}C(5+i,6)^p = Sum{i=0..6*(p-1)}b(6+i,6,p)*C(n+5+i,6*p+1) .

%F C(n,6)^p = Sum_{i=0..6*(p-1)}b(6+i,6,p)*C(n+i,6*p) .

%e For example :

%e b(6,6,p) = 1;

%e b(7,6,p) = 7^p - (6*p+1) ;

%e b(8,6,p) = 28^p - (6*p+1)*7^p + C(6*p+1,2) ;

%e b(9,6,p) = 84^p - (6*p+1)*28^p + C(6*p+1,2)*7^p + C(6*p+1,3) ;

%e b(10,6,p) = 210^p - (6*p+1)*84^p + C(6*p+1,2)*28^p - C(6*p+1,3)*7^p + C(6*p+1,4) .

%e Coefficients triangle:

%e 1;

%e 1, 36, 225, 400, 225, 36, 1;

%e 1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1;

%e 1, 2376, 554931, 35138736, 879018750, 10490842656, 66555527346, 239677178256, 509723668476, 654019630000, 509723668476, 239677178256, 66555527346, 10490842656, 879018750, 35138736, 554931, 2376, 1;

%e 1, 16776, 16689816, 3656408776, 286691702976, 10255094095176, 192698692565176, 2080037792142216, 13690633212385551, 57229721552316976, 156200093827061616, 283397584598631216, 345271537321293856, 283397584598631216, 156200093827061616, 57229721552316976,13690633212385551, 2080037792142216, 192698692565176, 10255094095176, 286691702976, 3656408776, 16689816, 16776, 1;

%e Example:

%e Sum_{i=1..n}C(5+i,6)^3 = C(n+6,19) + 324*C(n+7,19) + 15606*C(n+8,19) + 233300*C(n+9,19) +....

%e C(n,6)^3 = C(n,18) + 324*C(n+1,18) + 15606*C(n+2,18) + 233300*C(n+3,18)....

%t b[k_, 6, p_] := Sum[(-1)^i*Binomial[6*p+1, i]*Binomial[k-i, 6]^p /. k -> 6+i, {i, 0, k-6}]; row[p_] := Table[b[k, 6, p], {k, 6, 6*p}]; Table[row[p], {p, 1, 5}] // Flatten (* _Jean-François Alcover_, Feb 05 2014 *)

%Y Cf. A087127, A086023, A086024, A086025, A087107, A087108, A087109, A087110, A087111, A154283, A174266, A181544, A237202.

%K nonn,tabf

%O 1,3

%A _Yahia Kahloune_, Feb 05 2014

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Last modified April 10 06:23 EDT 2020. Contains 333392 sequences. (Running on oeis4.)