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

1,6

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).

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

Example

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;
  ...

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 *)

Prog

(Magma)
R<x>:=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
(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

Crossrefs

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

Sequence in context: A232186 A340061 A325757 * A350357 A136277 A133233

Adjacent sequences: A161158 A161159 A161160 * A161162 A161163 A161164

Keyword

easy,nonn,tabf

Author

Alford Arnold, Jun 04 2009

Status

approved