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%I #33 Mar 27 2022 03:47:33
%S 1,1,9,9,1,1,54,405,760,405,54,1,1,243,6750,49682,128124,128124,49682,
%T 6750,243,1,1,1008,83736,1722320,12750255,40241088,58571184,40241088,
%U 12750255,1722320,83736,1008,1,1,4077,922347,45699447,789300477,5904797049,21475242671,40396577931,40396577931,21475242671,5904797049,789300477,45699447,922347,4077,1
%N Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(3*n+1,i) * binomial(k+3-i,3)^n, 0 <= k <= 3*(n-1).
%C From _Yahia Kahloune_, Jan 30 2014: (Start)
%C In general, let 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,3,n).
%C With these coefficients we can calculate: Sum_{i=1..n} binomial(i+e-1,e)^p = Sum_{i=0..e*(p-1)} b(i,e,p)*binomial(n+e+i,e*p+1).
%C For example, A086020(n) = Sum_{i=1..n} binomial(2+i, 3)^2 = T(2,0)*binomial(n+3, 7) + T(2,1)*binomial(n+4,7) + T(2,2)*binomial(n+5,7) + T(2,3)*binomial(n+6,7) = (1/5040)*(20*n^7 + 210*n^6 + 854*n^5 + 1680*n^4 + 1610*n^3 + 630*n^2 + 36*n).
%C (End)
%C T(n,k) is the number of permutations of 3 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 Andrew Howroyd, <a href="/A174266/b174266.txt">Table of n, a(n) for n = 1..1335</a>
%F T(n,k) = [x^k] (1-x)^(3*n+1)*(Sum_{k>=0} (k*(k+1)*(k-1)/2)^n*x^k)/(3^n*x^2).
%F T(n,k) = T(n, 3*n-k).
%F From _Yahia Kahloune_, Jan 30 2014: (Start)
%F Sum_{i=1..n} binomial(2+i,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+3+i,3*p+1).
%F binomial(n,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+i,3*p). (End)
%e Triangle begins:
%e 1;
%e 1, 9, 9, 1;
%e 1, 54, 405, 760, 405, 54, 1;
%e 1, 243, 6750, 49682, 128124, 128124, 49682, ... ;
%e 1, 1008, 83736, 1722320, 12750255, 40241088, 58571184, ... ;
%e 1, 4077, 922347, 45699447, 789300477, ... ;
%e 1, 16362, 9639783, 1063783164, 38464072830, ... ;
%e 1, 65511, 98361900, 23119658500, 1641724670475, ... ;
%e 1, 262116, 992660346, 484099087156, 64856779908606, ... ;
%e ...
%e The T(2,1) = 9 permutations of 111222 with 1 descent are: 112221, 112212, 112122, 122211, 122112, 121122, 222111, 221112, 211122. - _Andrew Howroyd_, May 07 2020
%t (* First program *)
%t p[n_, x_]:= p[n,x]= (1-x)^(3*n+1)*Sum[(Binomial[k+1, 3])^n*x^k, {k, 0, Infinity}]/x^2;
%t Table[CoefficientList[p[x, n], x], {n,10}]//Flatten (* corrected by _G. C. Greubel_, Mar 26 2022 *)
%t (* Second program *)
%t T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j+1)*Binomial[3*n+1, k-j+1]*(j*(j^2-1)/2)^n, {j, 0, k+1}]/(3^n)];
%t Table[T[n, k], {n,10}, {k,3*n-2}]//Flatten (* _G. C. Greubel_, Mar 26 2022 *)
%o (PARI) T(n,k)={sum(i=0, k, (-1)^i*binomial(3*n+1, i)*binomial(k+3-i, 3)^n)} \\ _Andrew Howroyd_, May 06 2020
%o (Sage)
%o @CachedFunction
%o def T(n, k): return (1/3^n)*sum( (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(j^2-1)/2)^n for j in (0..k+1) )
%o flatten([[T(n, k) for k in (1..3*n-2)] for n in (1..10)]) # _G. C. Greubel_, Mar 26 2022
%Y Columns k=0..9 are A000012, A289254, A151632, A151633, A151634, A151635, A151636, A151637, A151638, A151639.
%Y Row sums are A014606.
%Y Similar triangles for e=1..6: A173018 (or A008292), A154283, this sequence, A236463, A237202, A237252.
%Y Cf. A060187, A174264.
%K nonn,tabf
%O 1,3
%A _Roger L. Bagula_, Mar 14 2010
%E Edited by _Andrew Howroyd_, May 06 2020
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