A022166 -id:A022166 - cp's OEIS frontend

A135950 Matrix inverse of triangle A022166.

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, -777240, 21590, -255, 1

Offset: 0

Paul D. Hanna, Dec 08 2007

A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.
The coefficient [x^k] of Product_{i=1..n} (x-2^(i-1)). - Roger L. Bagula, Mar 20 2009
Triangle T(n,k), 0 <= k <= n, read by rows given by (-1, 1-q, -q^2, q-q^3, -q^4, q^2-q^5, -q^6, q^3-q^7, -q^8, ...) DELTA (1, 0, q, 0, q^2, 0, q^3, 0, q^4, 0, ...) (for q = 2) = (-1, -1, -4, -6, -16, -28, -64, -120, -256, ...) DELTA (1, 0, 2, 0, 4, 0, 8, 0, 16, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013
Reversed rows of triangle A158474. - Werner Schulte, Apr 06 2019
T(n,k) = Sum mu(0,U) where the sum is taken over the subspaces U of GF(2)^n having dimension n-k and mu is the Moebius function of the poset of all subspaces of GF(2)^n. - Geoffrey Critzer, Jun 02 2024

			Triangle begins:
         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; ...

G. C. Greubel, Rows n = 0..75 of triangle, flattened

Cf. A022166, A006125, A028361, A127850, A135951 (central terms), A158474.

max = 9; M = Table[QBinomial[n, k, 2], {n, 0, max}, {k, 0, max}] // Inverse; Table[M[[n, k]], {n, 1, max+1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)
p[x_, n_, q_] := (-1)^n*q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, 2], {x, 0, n}], x], {n, 0, 10}]// Flatten (* G. C. Greubel, Apr 15 2019 *)

T(n,k)=local(q=2,A=matrix(n+1,n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1);A[n+1,k+1]

Unsigned column 0 equals A006125(n) = 2^(n*(n-1)/2).
Unsigned column 1 equals A127850(n) = (2^n-1)*2^(n*(n-1)/2)/2^(n-1).
Row sums equal 0^n.
Unsigned row sums equal A028361(n) = Product_{k=0..n} (1+2^k).
T(n,k) = (-1)^(n-k) * A022166(n,k) * 2^binomial(n-k,2) for 0 <= k <= n. - Werner Schulte, Apr 06 2019 [corrected by Werner Schulte, Dec 27 2021]
Sum_{n>=0} Sum_{k=0..n} T(n,k)y^k*x^n/A005329(n) = e(y*x)/e(x) where e(x) = Sum_{n>=0} x^n/A005329(n). - Geoffrey Critzer, Jun 02 2024

A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.

Original entry on oeis.org

1, 1, 2, 6, 28, 204, 2344, 43160, 1291952, 63647664, 5218320672, 719221578080, 168115994031040, 67159892835119296, 46166133463916209792, 54941957091151982047616, 113826217192695041078973184, 412563248965919999955196308224, 2627807814905396804499456018866688

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			The recurrence a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) is illustrated by:
a(2) = 1*(1) + 3*(1) + 1*(2) = 6;
a(3) = 1*(1) + 7*(1) + 7*(2) + 1*(6) = 28;
a(4) = 1*(1) + 15*(1) + 35*(2) + 15*(6) + 1*(28) = 204.
Triangle A022166 begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
1, 31, 155, 155, 31, 1;
1, 63, 651, 1395, 651, 63, 1; ...

Alois P. Heinz, Table of n, a(n) for n = 0..90

Cf. A022166, A125810, A125811, A125813, A125814, A125815.

Column k=2 of A381369.

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(2^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

a[0] = 1; a[n_] := a[n] = Sum[QBinomial[n-1, k, 2] a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

/* q-Binomial coefficients: */ {C_q(n,k)=if(n

a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 2^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.

Original entry on oeis.org

1, 2, 11, 100, 1677, 49974, 2801567, 293257480, 59426801521, 23154622451162, 17786849024835651, 26694462878992491180, 79786045619298591331605, 469805503062346255040726910, 5538428985758278544518994721255, 129179377104085570277109465712798800, 6048537751321912538368011648067930447545

Offset: 0

Paul D. Hanna, Jun 21 2014

nonn

a(n) is the number of Green's H classes in the semigroup of n X n matrices over GF(2) (cf. A359313). - Geoffrey Critzer, Jun 20 2023

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + ...
Related integer series:
A(x)^(1/2) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 + 145101805*x^7 + 29560055405*x^8 + ... + A243951(n)*x^n + ...
A022166, the triangle of q-binomial coefficients for q=2, begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   35,    15,     1;
  1,  31,  155,   155,    31,    1;
  1,  63,  651,  1395,   651,   63,   1;
  1, 127, 2667, 11811, 11811, 2667, 127, 1; ...
from which we can illustrate the initial terms of this sequence:
  a(0) = 1^2 = 1;
  a(1) = 1^2 + 1^2 = 2;
  a(2) = 1^2 + 3^2 + 1^2 = 11;
  a(3) = 1^2 + 7^2 + 7^2 + 1^2 = 100;
  a(4) = 1^2 + 15^2 + 35^2 + 15^2 + 1^2 = 1677;
  a(5) = 1^2 + 31^2 + 155^2 + 155^2 + 31^2 + 1^2 = 49974;
  a(6) = 1^2 + 63^2 + 651^2 + 1395^2 + 651^2 + 63^2 + 1^2 = 2801567; ...

Paul D. Hanna, Table of n, a(n) for n = 0..60
Wikipedia, Green's relations

Cf. A022166, A243951, A359313.

a[n_] := Sum[QBinomial[n, k, 2]^2, {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

{A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
{a(n)=sum(k=0,n,A022166(n, k)^2)}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * 2^(n^2/2), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd. - Vaclav Kotesovec, Jun 23 2014

Sum_{n>=0} a(n)*x^n/A005329(n)^2 = E(x)^2 where E(x) = Sum_{n>=0} x^n/A005329(n)^2. - Geoffrey Critzer, Jun 20 2023

A135951 Central terms of triangle A135950, the matrix inverse of triangle A022166.

Original entry on oeis.org

1, -3, 70, -11160, 12850368, -111842970624, 7558738517524480, -4024873276683363287040, 17013427111087951089139449856, -573105858480900876266937950612226048, 154142404695090288939416498797330749299097600

Offset: 0

Paul D. Hanna, Dec 08 2007

sign

A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.

Cf. A135950, A022166.

max = 20; M = Table[QBinomial[n, k, 2], {n, 0, max}, {k, 0, max}] // Inverse; Table[M[[n, (n+1)/2]], {n, 1, max+1, 2}] (* Jean-François Alcover, Apr 09 2016 *)
Table[(-1)^n 2^((n-1)n/2) QBinomial[2n, n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

a(n)=local(q=2,A=matrix(2*n+1,2*n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1);A[2*n+1,n+1]

A243951 Self-convolution square-root of A243950, which is the sums of the squares of the q-binomial coefficients for q=2 in rows of triangle A022166.

Original entry on oeis.org

1, 1, 5, 45, 781, 23981, 1371885, 145101805, 29560055405, 11546945197165, 8881721878376045, 13338290506465706605, 39879639563413780322925, 234862804790553590007179885, 2768979430068663216466330446445, 64586918396493458414460474344516205, 3024204274887062319005574660727125346925

Offset: 0

Paul D. Hanna, Jun 21 2014

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 +...
where
A(x)^2 = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + 293257480*x^7 + 59426801521*x^8 +...+ A243950(n)*x^n +...
The terms in this sequence appear to be divisible by 5 everywhere except
a(n) == 1 (mod 5) when n = {0,1,4,5,20,21,24,25,100,101,104,105,120,121,124, 125,500,501,...}.

Paul D. Hanna, Table of n, a(n) for n = 0..80

Cf. A243950, A022166.

a[n_] := SeriesCoefficient[Sqrt[Sum[x^m Sum[QBinomial[m, k, 2]^2, {k, 0, m}], {m, 0, n}]], {x, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

{A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
{a(n)=polcoeff(sqrt(sum(m=0,n,x^m*sum(k=0,m,A022166(m, k)^2) +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * 2^(n^2/2-1), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd (constants same as for A243950). - Vaclav Kotesovec, Jun 23 2014

A143774 Eigentriangle of triangle A022166.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 14, 6, 1, 15, 70, 70, 28, 1, 31, 310, 930, 868, 204, 1, 63, 1302, 8370, 18228, 12852, 2344

Offset: 0

Gary W. Adamson, Aug 31 2008

An eigentriangle of triangle T may be defined by taking the termwise product of row n-1 of T and the first n terms of the eigensequence of T; 0<=k<=n.
Row sums = A125812 shifted 1 place to the left: (1, 2, 6, 28, 204,...).
Sum of n-th row terms = rightmost term of (n+1)-th row.
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
... (and the eigensequence of A022166 = A125812: (1, 1, 2, 6, 28, 204,...) we apply the termwise product of (n-1)-th row of A022166 and the first n terms of A125812.

			First few rows of the triangle:
  1;
  1, 1;
  1, 3, 2;
  1, 7, 14, 6;
  1, 15, 70, 90, 28;
  1, 31, 310, 930, 868, 204;
  ...
Row 3 of A022166 = (1, 7, 7, 1), first 4 terms of A143774 = (1, 1, 2, 6), so row 3 of A143774 = (1*1, 7*1, 7*2, 1*6) = (1, 7, 14, 6).

Cf. A022166, A125812.

Given triangle A022166: 1;

A156823 Triangle T(n,k,2) read by rows (generalized q-Stirling numbers of second kind): T(n, k, q_) = (1/(q - 1)^k)Sum[(-1)^(k - j)q*Binomial[k + n, k -j] - Binomial[j + n, j, q - 1], {j, 0, k}], with q=2, where Binomial[,] is the Gaussian q-binomial coefficient as in A022166.

Original entry on oeis.org

1, 1, 1, 1, 4, 13, 1, 11, 90, 670, 1, 26, 480, 7870, 122861, 1, 57, 2247, 77527, 2526198, 80189094, 1, 120, 9807, 695368, 46334382, 2999255160, 191467330714, 1, 247, 41176, 5924720, 798773822, 104443530554, 13455795711072, 1721026866650520, 1

Offset: 0

Roger L. Bagula, Feb 16 2009

Row sums are 1, 2, 18, 772, 131238, 82795124, 194513625552, 1734587910632112, 59780354709947486310, 8067711354683582659357588, 4300494571012469622746969756172,....

			Triangle begins:
{1},
{1, 1},
{1, 4, 13},
{1, 11, 90, 670},
{1, 26, 480, 7870, 122861},
{1, 57, 2247, 77527, 2526198, 80189094},
{1, 120, 9807, 695368, 46334382, 2999255160, 191467330714},
{1, 247, 41176, 5924720, 798773822, 104443530554, 13455795711072, 1721026866650520},
{1, 502, 169186, 49067150, 13310897072, 3498722283914, 905629978109142, 232656671284481730, 59546788896602477613},
{1, 1013, 686829, 400036769, 217729686031, 114758591845755, 59547270411289947, 30661311851453644647, 15727477144989414892230, 8051953156564494657274366},
{1, 2036, 2769657, 3233395880, 3525493671271, 3721338617555988, 3866476676171065671, 3986066951574453826080, 4093473968605655678972070, 4195675823040150254245701976, 4296294797725523713719072795542}
...

T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russian Journal of Mathematical Physics, Volume 15, Number 1, March 2008, pp. 51-57, DOI:10.1134/S1061920808010068.

Cf. A022166.

t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
t1[n_, k_, q_] = (1/(q - 1)^k)*Sum[(-1)^(k - j)* Binomial[k + n, k - j]*b[j + n, j, q - 1], {j, 0, k}];
Table[Flatten[Table[Table[t1[n, k, m + 1], {k, 0, n}], {n, 0, 10}]], {m, 1, 15}]

t1(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}]; q=2; m=1.

A174526 Triangle t(n,m) = 2*A022166(n,m)-binomial(n,m), read by rows, 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 64, 26, 1, 1, 57, 300, 300, 57, 1, 1, 120, 1287, 2770, 1287, 120, 1, 1, 247, 5313, 23587, 23587, 5313, 247, 1, 1, 502, 21562, 194254, 401504, 194254, 21562, 502, 1, 1, 1013, 86834, 1575986, 6619368, 6619368, 1575986, 86834

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 21 2010

Row sums are 1, 2, 6, 24, 118, 716, 5586, 58296, 834142, 16566404, 459510186,... = 2*A006116(n)-2^n.

The column m=1 might be A000295. - R. J. Mathar, Nov 14 2011

			1;
1, 1;
1, 4, 1;
1, 11, 11, 1;
1, 26, 64, 26, 1;
1, 57, 300, 300, 57, 1;
1, 120, 1287, 2770, 1287, 120, 1;
1, 247, 5313, 23587, 23587, 5313, 247, 1;
1, 502, 21562, 194254, 401504, 194254, 21562, 502, 1;
1, 1013, 86834, 1575986, 6619368, 6619368, 1575986, 86834, 1013, 1;
1, 2036, 348457, 12695310, 107487764, 218443050, 107487764, 12695310, 348457, 2036, 1;

Cf. A008292.

A174526 := proc(n,k)
   2*A022166(n,k)-binomial(n,k) ;
end proc:
seq(seq(A174526(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] = 2*c[n, q]/(c[m, q]*c[n - m, q]) - Binomial[n, m];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
(* second program: *)
t[n_, m_] := 2 QBinomial[n, m, 2] - Binomial[n, m]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

A156824 Generalized q-Stirling 2nd numbers (see A022166):q=3;m=2; t1(n, k, q_) = (1/(q - 1)^k)Sum[(-1)^(k - j)Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}].

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 1, 18, 255, 3400, 1, 58, 2575, 106400, 4300541, 1, 179, 24234, 3038714, 371984935, 45182779173, 1, 543, 221886, 83805218, 30877084287, 11284441459641, 4113010719221412, 1, 1636, 2010034, 2280772380, 2523761295627, 2769755537579952, 3031455813294108948, 3314879002466198503080

Offset: 0

Roger L. Bagula, Feb 16 2009

Row sums are: {1, 2, 27, 3674, 4409575, 45557827236, 4124326121792988, 3317913230561074271658, 23891408190421363405102296351, 1544865931069396100350109616919010834, 898255701914264060744770399113246348926078875,...}.

			{1},
{1, 1},
{1, 5, 21},
{1, 18, 255, 3400},
{1, 58, 2575, 106400, 4300541},
{1, 179, 24234, 3038714, 371984935, 45182779173},
{1, 543, 221886, 83805218, 30877084287, 11284441459641, 4113010719221412},
{1, 1636, 2010034, 2280772380, 2523761295627, 2769755537579952, 3031455813294108948, 3314879002466198503080},
{1, 4916, 18134514, 61761978300, 205103050119627, 675507759929956512, 2218696908383551468308, 7280640738500515014553320, 23884125330310241581776080853},
{1, 14757, 163358151, 1669369542291, 16633368715805358, 164364489292170484590, 1619729636032633290318498, 15947039988935644725038892138, 156958704656445989980689513610911, 1544708956416079771407327238656984139},
{1, 44281, 1470710395, 45090623244271, 1347888929379662362, 39959437240297322060278, 1181382154718570769797966170, 34895073775900019052240192095218, 1030399116864328608488320120932827143, 30423048235258853916780570577659445554071, 898225277835594788771326994531551239761914685}

T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russian Journal of Mathematical Physics, Volume 15, Number 1, March 2008, pp. 51-57.

Cf. A022166.

t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
t1[n_, k_, q_] = (1/(q - 1)^k)*Sum[(-1)^(k - j)* Binomial[k + n, k - j]*b[j + n, j, q - 1], {j, 0, k}];
Table[Flatten[Table[Table[t1[n, k, m + 1], {k, 0, n}], {n, 0, 10}]], {m, 1, 15}]
(* Second program: *)
(* S stands 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;
Table[S[n+k, n, 3], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 09 2020 *)

t1(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}]; q=3;m=2.

A156825 Generalized q-Stirling 2nd numbers (see A022166):q=4;m=3; t1(n, k, q_) = (1/(q - 1)^k)Sum[(-1)^(k - j)Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}].

Original entry on oeis.org

1, 1, 1, 1, 6, 31, 1, 27, 598, 12714, 1, 112, 10118, 872744, 74451015, 1, 453, 164591, 56998275, 19510862790, 6659538174846, 1, 1818, 2646161, 3669008040, 5027706837390, 6869479371212196, 9379110782727354118, 1, 7279, 42396780

Offset: 0

Roger L. Bagula, Feb 16 2009

Row sums are: {1, 2, 38, 13340, 75333990, 6679106200956, 9385985293477059724, 210307101689444749681505920, 75309752513141244017422009494610310, 431334730561934365895986795984802627076981452, 39523158749221869286186846414773795221687625241015791028,...}.

			{1},
{1, 1},
{1, 6, 31},
{1, 27, 598, 12714},
{1, 112, 10118, 872744, 74451015},
{1, 453, 164591, 56998275, 19510862790, 6659538174846},
{1, 1818, 2646161, 3669008040, 5027706837390, 6869479371212196, 9379110782727354118},
{1, 7279, 42396780, 235197823620, 1289443021626210, 7048517820471945006, 38501334928380019031884, 210268593304708870928675140},
{1, 29124, 678610560, 15059445506820, 330263030118109110, 7221644410750565452956, 157795323487774482338855704, 3447249110183738275563231529020, 75306305106228514816683123116517015},
{1, 116505, 10858933965, 963923954302485, 84558902081023550895, 7396063067152669466208951, 646433182194355185109143203035, 56489425142435134168297605455930355, 4936177764676230687274829745467766867270, 431329794327679618082286045004555759407867990},
{1, 466030, 173748069715, 61693218021437860, 21647880931024091567395, 7573871645483348274559946326, 2647903920069761660850674382798185, 925565107087525879643000261252991542480, 323513080232532159312906941144197336652189270, 113076340698070130663461880889470578649115862464740, 39523045672557657210255895794560156111877694170705309026}

T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russian Journal of Mathematical Physics, Volume 15, Number 1, March 2008, pp. 51-57, DOI:10.1134/S1061920808010068.

Cf. A022166.

t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
t1[n_, k_, q_] = (1/(q - 1)^k)*Sum[(-1)^(k - j)* Binomial[k + n, k - j]*b[j + n, j, q - 1], {j, 0, k}];
Table[Flatten[Table[Table[t1[n, k, m + 1], {k, 0, n}], {n, 0, 10}]], {m, 1, 15}]

t1(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}]; q=4;m=3.

cp's OEIS Frontend

A135950 Matrix inverse of triangle A022166.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A135951 Central terms of triangle A135950, the matrix inverse of triangle A022166.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A243951 Self-convolution square-root of A243950, which is the sums of the squares of the q-binomial coefficients for q=2 in rows of triangle A022166.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A143774 Eigentriangle of triangle A022166.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A156823 Triangle T(n,k,2) read by rows (generalized q-Stirling numbers of second kind): T(n, k, q_) = (1/(q - 1)^k)*Sum[(-1)^(k - j)*q*Binomial[k + n, k -j] - Binomial[j + n, j, q - 1], {j, 0, k}], with q=2, where Binomial[,] is the Gaussian q-binomial coefficient as in A022166.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A174526 Triangle t(n,m) = 2*A022166(n,m)-binomial(n,m), read by rows, 0<=m<=n.

Views

Author

Keywords

A156823 Triangle T(n,k,2) read by rows (generalized q-Stirling numbers of second kind): T(n, k, q_) = (1/(q - 1)^k)Sum[(-1)^(k - j)q*Binomial[k + n, k -j] - Binomial[j + n, j, q - 1], {j, 0, k}], with q=2, where Binomial[,] is the Gaussian q-binomial coefficient as in A022166.

A156824 Generalized q-Stirling 2nd numbers (see A022166):q=3;m=2; t1(n, k, q_) = (1/(q - 1)^k)Sum[(-1)^(k - j)Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}].

A156825 Generalized q-Stirling 2nd numbers (see A022166):q=4;m=3; t1(n, k, q_) = (1/(q - 1)^k)Sum[(-1)^(k - j)Binomial[k + n, k -j]*q-Binomial[j + n, j, q - 1], {j, 0, k}].