A308326 The q-analog T(q; n,k) of the triangle A163626 for 0 <= k <= n, for q = 2.

1, 1, -1, 1, -4, 3, 1, -13, 33, -21, 1, -40, 270, -546, 315, 1, -121, 2010, -10080, 17955, -9765, 1, -364, 14433, -165270, 707805, -1171800, 615195, 1, -1093, 102123, -2580081, 24421005, -95765355, 151953165, -78129765, 1, -3280, 718140, -39416076, 795752370, -6790268520, 25331269320, -39221142030, 19923090075

Offset

0,5

Comments

The formulas are given for the general case depending on some fixed integer q. The terms are valid if q = 2.

Special cases: T(0; n,k) = (-1)^k * binomial(n,k) for 0 <= k <= n and T(1; n,k) = A163626(n,k) for 0 <= k <= n.

Links

Table of n, a(n) for n=0..44.

Formula

T(q; n,k) = [k+1]_q * T(q; n-1,k) - [k]_q * T(q; n-1,k-1) for 1 <= k <= n with initial values T(q; n,0) = 1 for n >= 0 and T(q; i,j) = 0 if i < j or j < 0 where [i]_q = (q^i - 1)/(q - 1) for i >= 0.

T(q; n,k) = (1/q^binomial(k+1,2)) * (Sum_{j=0..k} (-1)^j * [k,j]_q * q^binomial(k-j,2) * ([j+1]_q)^n) for 0 <= k <= n and q not equal zero where [m,i]_q are the q-binomials (here A022166 for q = 2) and [i]_q = (q^i - 1)/(q - 1) for i >= 0.

Sum_{k=0..n} T(q; n,k) = A000007(n) for n >= 0.

T(q; n,k)/T(q; k,k) give the q-analogs of the Stirling numbers of the second kind (for q = 2 see A139382, but offset 1).

T(q; n,n) = (-1)^n * Product_{j=1..n} [j]_q for n>=0 with empty product 1 (case n = 0) where [i]_q = (q^i - 1)/(q - 1) for i >= 0.

T(q; n,1) = -[n,1]_(q+1) for n >= 1 where [m,i]_q are the q-binomials (here A022166 for q = 2 and A022167 for q = 3).

G.f. of column k: col(q; t,k) = Sum_{n>=k} T(q; n,k)*t^n = ((-t)^k/(1-t)) * Product_{j=1..k} ([i]_q/(1-[i+1]_q*t)) for k>=0 with empty product 1 (case k=0) and [i]_q = i if q = 1 otherwise (q^i-1)/(q-1) for i>=0.

Example

If q = 2 the triangle T(2; n,k) starts:
n\k:  0     1      2        3        4         5         6         7
====================================================================
  0:  1
  1:  1    -1
  2:  1    -4      3
  3:  1   -13     33      -21
  4:  1   -40    270     -546      315
  5:  1  -121   2010   -10080    17955     -9765
  6:  1  -364  14433  -165270   707805  -1171800    615195
  7:  1 -1093 102123 -2580081 24421005 -95765355 151953165 -78129765
etc.

Prog

(PARI) q = 2; {T(n, k) = if(k<0 || k>n, 0, if(k==0, 1, if(q==1, (k+1) * T(n-1, k) - k * T(n-1, k-1), ((q^(k+1) - 1)/(q - 1)) * T(n-1, k) - ((q^k - 1)/(q - 1)) * T(n-1, k-1))))};
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))) \\ Werner Schulte, May 26 2019

Crossrefs

Cf. A000007, A007318, A022166, A022167, A139382, A163626.

Sequence in context: A112493 A370609 A010305 * A098234 A193795 A181355

Adjacent sequences: A308323 A308324 A308325 * A308327 A308328 A308329

Keyword

sign,tabl

Author

Werner Schulte, May 23 2019

Status

approved