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

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, 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, 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

Offset: 1

Author

Alford Arnold, Jun 04 2009

Keywords

• easy
• nonn
• tabf

Examples

			The differences between 5 3 4 3 1 and 4 2 2 yield row four : 1 1 2 3 1.
Triangle begins:
  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, 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, 11, 15, 22, 21, 25, 25, 27, 23, 22, 15, 13, 8, 6, 2, 2;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..2390 (rows 1..30, flattened)
• William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 16.
• M. Isachenkov, I. Kirsch, and V. Schomerus, Chiral Primaries in Strange Metals, arXiv preprint arXiv:1403.6857 [hep-th], 2014. See Eq. (4.6).

Crossrefs

Cf. A000079 (row sums), A002865 (antidiagonal sums), A077957 (alternating row sums).

Programs

Magma

R:=PowerSeriesRing(Rationals(), 100);
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]]) >;
A083906:= func< n,k | Coefficient(R!( (&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
A161161:= func< n,k | A083906(n,k) - A083906(n-1,k) >;
[A161161(n,k): k in [0..Floor(n^2/4)], n in [1..12]]; // G. C. Greubel, Feb 13 2024

Maple

A161161 := proc(n,m)
     A083906(n,m)-A083906(n-1,m) ;
end proc:
for n from 0 to 10 do
     for k from 0 to A033638(n)-1 do
         printf("%d, ", A161161(n, k)) ;
     od:
od: # R. J. Mathar, Jul 13 2012

Mathematica

b[n_, k_] := b[n, k] = SeriesCoefficient[Sum[QBinomial[n, m, q], {m, 0, n}], {q, 0, k}];
T[n_, k_] := b[n, k] - b[n - 1, k];
Table[Table[T[n, k], {k, 0, n^2/4}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

SageMath

def t(n, k): # t = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*t(n-1, k) - t(n-2, k) + t(n-2, k-n+1)
def A161161(n,k): return t(n, k) - t(n-1, k)
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

Formula

Sum_{k=0..floor(n^2/4)} T(n, k) = A000079(n-1) (row sums).

Sum_{k=0..(n+2 - ceiling(sqrt(4*n)))} T(n-k, k) = A002865(n+1) (antidiagonal sums).

Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A077957(n-1). - G. C. Greubel, Feb 13 2024