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%I #28 May 07 2020 22:55:21
%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 10490842656,66555527346,239677178256,509723668476,654019630000,509723668476,239677178256,66555527346,10490842656,879018750,35138736,554931,2376,1
%N Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(6*n+1,i) * binomial(k+6-i,6)^n, 0 <= k <= 6*(n-1).
%C In general, define b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,6,n).
%C Using these coefficients we can obtain formulas for binomial(n,e)^p and for Sum_{i=1..n} binomial(e-1+i,e)^p.
%C In particular:
%C binomial(n, e)^p = Sum_{k=0..e*(p-1)} b(k,e p) * binomial(n+k, e*p).
%C Sum_{i=1..n} binomial(e-1+i, e)^p = Sum_{k=0..e*(p-1)} b(k,e,p) * binomial(n+e+k, e*p+1).
%C T(n,k) is the number of permutations of 6 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - _Andrew Howroyd_, May 06 2020
%H G. C. Greubel, <a href="/A237252/b237252.txt">Table of n, a(n) for the first 25 rows, flattened</a>
%F Sum_{i=1..n} binomial(5+i,6)^p = Sum{k=0..6*(p-1)} T(p,k) * binomial(n+6+k, 6*p+1).
%F binomial(n,6)^p = Sum_{k=0..6*(p-1)} T(p,k) * binomial(n+k, 6*p).
%e For example :
%e T(n,0) = 1;
%e T(n,1) = 7^n - (6*n+1);
%e T(n,2) = 28^n - (6*n+1)*7^n + C(6*n+1,2);
%e T(n,3) = 84^n - (6*n+1)*28^n + C(6*n+1,2)*7^n + C(6*n+1,3);
%e T(n,4) = 210^n - (6*n+1)*84^n + C(6*n+1,2)*28^n - C(6*n+1,3)*7^n + C(6*n+1,4).
%e Triangle T(n,k) begins:
%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 ...
%e Example:
%e Sum_{i=1..n} C(5+i,6)^2 = A086027(n) = C(n+6,13) + 36*C(n+7,13) + 225*C(n+8,13) + 400*C(n+9,13) + 225*C(n+10,13) + 36*C(n+11,13) + C(n+12,13).
%e binomial(n,6)^2 = C(n,12) + 36*C(n+1,12) + 225*C(n+2,12) + 400*C(n+3,12) + 225*C(n+4,12) + 36*C(n+5,12) + C(n+6,12).
%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 *)
%o (PARI) T(n,k)={sum(i=0, k, (-1)^i*binomial(6*n+1, i)*binomial(k+6-i, 6)^n)} \\ _Andrew Howroyd_, May 06 2020
%Y Columns k=2..6 are A151651, A151652, A151653, A151654, A151655.
%Y Row sums are A248814.
%Y Similar triangles for e=1..5: A173018 (or A008292), A154283, A174266, A236463, A237202.
%Y Sum_{i=1..n} binomial(5+i,6)^p for p=1..3 gives: A000580, A086027, A086028.
%Y Cf. A087127, A086023, A086024, A086025, A087107, A087108, A087109, A087110, A087111, A181544.
%K nonn,tabf
%O 1,3
%A _Yahia Kahloune_, Feb 05 2014
%E Edited by _Andrew Howroyd_, May 06 2020
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