A157783 -id:A157783 - cp's OEIS frontend

A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 130, 40, 1, 1, 121, 1210, 1210, 121, 1, 1, 364, 11011, 33880, 11011, 364, 1, 1, 1093, 99463, 925771, 925771, 99463, 1093, 1, 1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157783(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    4,      1;
  1,   13,     13,        1;
  1,   40,    130,       40,        1;
  1,  121,   1210,     1210,      121,        1;
  1,  364,  11011,    33880,    11011,      364,      1;
  1, 1093,  99463,   925771,   925771,    99463,   1093,    1;
  1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Rows n = 0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for sequences related to Gaussian binomial coefficients

Columns k=0..3 give A000012, A003462, A006100, A006101.

Cf. A006117 (row sums).

A022167 := proc(n,m)
        A027871(n)/A027871(n-m)/A027871(m) ;
end proc:
seq(seq(A022167(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

p[n_] := Product[3^k-1, {k, 1, n}]; t[n_, m_] := p[n]/(p[n-m]*p[m]); Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014, after R. J. Mathar *)
Table[QBinomial[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten
(* or, after Vladimir Kruchinin, using S for qStirling2: *)
S[n_, k_, q_] /; 1 <= k <= n := S[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}]* S[n-1, k, q]; S[n_, 0, ] := KroneckerDelta[n, 0]; S[0, k, ] := KroneckerDelta[0, k]; S[, , ] = 0;
T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n-j, n-k, q]*(q-1)^(k-j) /. q -> 3, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020 *)

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
T(n,k) = Sum_{j=0..k} C(n,j)*qStirling2(n-j,n-k,3)*(2)^(k-j),j,0,k), n >= k, where qStirling2(n,k,3) is triangle A333143. - Vladimir Kruchinin, Mar 07 2020
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 1. - Seiichi Manyama, May 09 2025

A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 5, -6, 1, 125, -155, 31, -1, 15625, -19500, 4030, -156, 1, 9765625, -12203125, 2538250, -101530, 781, -1, 30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1, 476837158203125, -596038818359375

Offset: 0

Roger L. Bagula, Mar 07 2009

Except for n=0, the row sums are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=5)= [1,4,25,120,625,3100,15625,...] DELTA [ -1,0,-5,0,-25,0,-125,0,-625,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  5, -6, 1;
  125, -155, 31, -1;
  15625, -19500, 4030, -156, 1;
  9765625, -12203125, 2538250, -101530, 781, -1;
  30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1;
  476837158203125, -596038818359375, 124166806640625, -5005123921875, 40040991375, -63573405, 19531, -1;

Cf. A135950, A157783, A109345 (first column), A003463 (first subdiagonal).

A157832 := proc(n,k)
        product( 5^(i-1)-x,i=1..n) ;
        coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

p[x_, n_] = If[n == 0, 1, Product[q^(i - 1) - x, {i, 1, n}]];
q = 5;
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 4, -5, 1, 64, -84, 21, -1, 4096, -5440, 1428, -85, 1, 1048576, -1396736, 371008, -23188, 341, -1, 1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1, 4398046511104, -5863704100864, 1563272675328, -99158478848

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.
The matrix inverses seem to be related to the Gaussian q-form combinations.
Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=4)=[1,3,16,60,256,1008,4096,16320,65536,261888,...] DELTA [ -1,0,-4,0,-16,0,-64,0,-256,0,-1024,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  4, -5, 1;
  64, -84, 21, -1;
  4096, -5440, 1428, -85, 1;
  1048576, -1396736, 371008, -23188, 341, -1;
  1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1;
  4398046511104, -5863704100864, 1563272675328, -99158478848, 1549351232, -5963412, 5461, -1;
  72057594037927936, -96075326035066880, 25618523216674816, -1626175790120960, 25483729063936, -99253893440, 95436436, -21845, 1;
Row n=3 represents 64 - 84*x + 21*x^2 - x^3.

Cf. A135950, A022166, A157783, A157832.

A157784 := proc(n,k)
    product( 4^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 4;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

A157963 Triangle T(n,k), 0<=k<=n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,-q^5,0,...] (for q=2) = [1,1,4,6,16,28,64,...] DELTA [ -1,0,-2,0,-4,0,-8,0,-16,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, -1, 2, -3, 1, 8, -14, 7, -1, 64, -120, 70, -15, 1, 1024, -1984, 1240, -310, 31, -1, 32768, -64512, 41664, -11160, 1302, -63, 1, 2097152, -4161536, 2731008, -755904, 94488, -5334, 127, -1, 268435456, -534773760, 353730560, -99486720, 12850368

Offset: 0

Philippe Deléham, Mar 10 2009

Row sums equal 0^n.
Row n contains the coefficients of Product_{j=0..n-1} (2^j*x-1), highest coefficient first. - Alois P. Heinz, Mar 26 2012
The elements of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^k*A022166(n,k). - R. J. Mathar, Mar 26 2013

			Triangle begins :
1;
1,    -1;
2,    -3,  1;
8,   -14,  7,  -1;
64, -120, 70, -15,  1;

Alois P. Heinz, Rows n = 0..44

Cf. A007179, A130595, A157783, A157784, A157785, A157832, A158474.

T:= n-> seq (coeff (mul (2^j*x-1, j=0..n-1), x, n-k), k=0..n):
seq (T(n), n=0..10);  # Alois P. Heinz, Mar 26 2012

row[n_] := CoefficientList[(-1)^n QPochhammer[x, 2, n] + O[x]^(n+1), x] // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 26 2016 *)

T(n,k) = (-1)^n*A135950(n,k). T(n,0) = A006125(n).

T(n,k) = [x^(n-k)] Product_{j=0..n-1} (2^j*x-1). - Alois P. Heinz, Mar 26 2012

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A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

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A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

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A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.

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A157963 Triangle T(n,k), 0<=k<=n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,-q^5,0,...] (for q=2) = [1,1,4,6,16,28,64,...] DELTA [ -1,0,-2,0,-4,0,-8,0,-16,0,...] where DELTA is the operator defined in A084938.

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Examples

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