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A161161 Irregular triangle of differences T(n,k) = A083906(n,k) - A083906(n-1,k) of q-Binomial coefficients.

%I A161161 #32 Feb 18 2024 01:38:07
%S A161161 1,1,1,1,1,2,1,1,2,3,1,1,1,2,3,5,2,2,1,1,2,3,5,7,5,4,3,1,1,1,2,3,5,7,
%T A161161 11,8,9,7,6,2,2,1,1,2,3,5,7,11,15,14,15,15,13,11,7,4,3,1,1,1,2,3,5,7,
%U A161161 11,15,22,21,25,25,27,23,22,15,13,8,6,2,2,1,1,2,3,5,7,11,15,22,30,32,37,42,44
%N A161161 Irregular triangle of differences T(n,k) = A083906(n,k) - A083906(n-1,k) of q-Binomial coefficients.
%H A161161 Michael De Vlieger, <a href="/A161161/b161161.txt">Table of n, a(n) for n = 1..2390</a> (rows 1..30, flattened)
%H A161161 William Q. Erickson and Jan Kretschmann, <a href="https://arxiv.org/abs/2311.07522">The structure and normalized volume of Monge polytopes</a>, arXiv:2311.07522 [math.CO], 2023. See p. 16.
%H A161161 M. Isachenkov, I. Kirsch, and V. Schomerus, <a href="http://arxiv.org/abs/1403.6857">Chiral Primaries in Strange Metals</a>, arXiv preprint arXiv:1403.6857 [hep-th], 2014. See Eq. (4.6).
%F A161161 Sum_{k=0..floor(n^2/4)} T(n, k) = A000079(n-1) (row sums).
%F A161161 Sum_{k=0..(n+2 - ceiling(sqrt(4*n)))} T(n-k, k) = A002865(n+1) (antidiagonal sums).
%F A161161 Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A077957(n-1). - _G. C. Greubel_, Feb 13 2024
%e A161161 The differences between 5 3 4 3 1 and 4 2 2 yield row four : 1 1 2 3 1.
%e A161161 Triangle begins:
%e A161161   1;
%e A161161   1, 1;
%e A161161   1, 1, 2;
%e A161161   1, 1, 2, 3, 1;
%e A161161   1, 1, 2, 3, 5, 2,  2;
%e A161161   1, 1, 2, 3, 5, 7,  5,  4,  3,  1;
%e A161161   1, 1, 2, 3, 5, 7, 11,  8,  9,  7,  6,  2,  2;
%e A161161   1, 1, 2, 3, 5, 7, 11, 15, 14, 15, 15, 13, 11,  7,  4,  3,  1;
%e A161161   1, 1, 2, 3, 5, 7, 11, 15, 22, 21, 25, 25, 27, 23, 22, 15, 13, 8, 6, 2, 2;
%e A161161   ...
%p A161161 A161161 := proc(n,m)
%p A161161      A083906(n,m)-A083906(n-1,m) ;
%p A161161 end proc:
%p A161161 for n from 0 to 10 do
%p A161161      for k from 0 to A033638(n)-1 do
%p A161161          printf("%d, ", A161161(n, k)) ;
%p A161161      od:
%p A161161 od: # _R. J. Mathar_, Jul 13 2012
%t A161161 b[n_, k_] := b[n, k] = SeriesCoefficient[Sum[QBinomial[n, m, q], {m, 0, n}], {q, 0, k}];
%t A161161 T[n_, k_] := b[n, k] - b[n - 1, k];
%t A161161 Table[Table[T[n, k], {k, 0, n^2/4}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Nov 25 2017 *)
%o A161161 (Magma)
%o A161161 R<x>:=PowerSeriesRing(Rationals(), 100);
%o A161161 qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
%o A161161 A083906:= func< n,k | Coefficient(R!( (&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
%o A161161 A161161:= func< n,k | A083906(n,k) - A083906(n-1,k) >;
%o A161161 [A161161(n,k): k in [0..Floor(n^2/4)], n in [1..12]]; // _G. C. Greubel_, Feb 13 2024
%o A161161 (SageMath)
%o A161161 def t(n, k): # t = A083906
%o A161161     if k<0 or k> (n^2//4): return 0
%o A161161     elif n<2 : return n+1
%o A161161     else: return 2*t(n-1, k) - t(n-2, k) + t(n-2, k-n+1)
%o A161161 def A161161(n,k): return t(n, k) - t(n-1, k)
%o A161161 flatten([[A161161(n, k) for k in range(int(n^2//4)+1)] for n in range(1,13)]) # _G. C. Greubel_, Feb 13 2024
%Y A161161 Cf. A000079 (row sums), A002865 (antidiagonal sums), A077957 (alternating row sums).
%K A161161 easy,nonn,tabf
%O A161161 1,6
%A A161161 _Alford Arnold_, Jun 04 2009