This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174266 #49 Jun 11 2025 01:24:13
%S A174266 1,1,9,9,1,1,54,405,760,405,54,1,1,243,6750,49682,128124,128124,49682,
%T A174266 6750,243,1,1,1008,83736,1722320,12750255,40241088,58571184,40241088,
%U A174266 12750255,1722320,83736,1008,1,1,4077,922347,45699447,789300477,5904797049,21475242671,40396577931,40396577931,21475242671,5904797049,789300477,45699447,922347,4077,1
%N A174266 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 A174266 From _Yahia Kahloune_, Jan 30 2014: (Start)
%C A174266 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 A174266 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 A174266 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). (End)
%C A174266 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 A174266 Andrew Howroyd, <a href="/A174266/b174266.txt">Table of n, a(n) for n = 1..1335</a>
%H A174266 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 12.
%F A174266 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 A174266 T(n,k) = T(n, 3*n-k).
%F A174266 From _Yahia Kahloune_, Jan 30 2014: (Start)
%F A174266 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 A174266 binomial(n,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+i,3*p). (End)
%F A174266 From _Sergii Voloshyn_, Dec 18 2024: (Start)
%F A174266 Let E be the operator (x^2)D*(1/x)*D*(x^2)*D, where D denotes the derivative operator d/dx. Then (1/6^n)* E^n(x^2/(1 - x)^4) = (row n generating polynomial)/(1 - x)^(3*n+4) = Sum_{i=0..k} (-1)^i * binomial(3*n+1,i) * binomial(k+3-i,3)^n.
%F A174266 For example, when n = 3 we have 1/216*E^3(x^2/(1 - x)^4) = x^2 (1 + 243x + 6750x^2 + 49682x^3 + 128124x^4 + 128124x^5 + 49682x^6 + 6750x^7 + 243x^8 + x^9)/(1 - x)^13. (End)
%e A174266 Triangle begins:
%e A174266 1;
%e A174266 1, 9, 9, 1;
%e A174266 1, 54, 405, 760, 405, 54, 1;
%e A174266 1, 243, 6750, 49682, 128124, 128124, 49682, ... ;
%e A174266 1, 1008, 83736, 1722320, 12750255, 40241088, 58571184, ... ;
%e A174266 1, 4077, 922347, 45699447, 789300477, ... ;
%e A174266 1, 16362, 9639783, 1063783164, 38464072830, ... ;
%e A174266 1, 65511, 98361900, 23119658500, 1641724670475, ... ;
%e A174266 1, 262116, 992660346, 484099087156, 64856779908606, ... ;
%e A174266 ...
%e A174266 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 A174266 (* First program *)
%t A174266 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 A174266 Table[CoefficientList[p[x, n], x], {n,10}]//Flatten (* corrected by _G. C. Greubel_, Mar 26 2022 *)
%t A174266 (* Second program *)
%t A174266 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 A174266 Table[T[n, k], {n,10}, {k,3*n-2}]//Flatten (* _G. C. Greubel_, Mar 26 2022 *)
%o A174266 (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 A174266 (Sage)
%o A174266 @CachedFunction
%o A174266 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 A174266 flatten([[T(n, k) for k in (1..3*n-2)] for n in (1..10)]) # _G. C. Greubel_, Mar 26 2022
%Y A174266 Columns k=0..9 are A000012, A289254, A151632, A151633, A151634, A151635, A151636, A151637, A151638, A151639.
%Y A174266 Row sums are A014606.
%Y A174266 Similar triangles for e=1..6: A173018 (or A008292), A154283, this sequence, A236463, A237202, A237252.
%Y A174266 Cf. A060187, A174264.
%K A174266 nonn,tabf
%O A174266 1,3
%A A174266 _Roger L. Bagula_, Mar 14 2010
%E A174266 Edited by _Andrew Howroyd_, May 06 2020