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A308326 The q-analog T(q; n,k) of the triangle A163626 for 0 <= k <= n, for q = 2. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A112493 A370609 A010305 * A098234 A193795 A181355
KEYWORD
sign,tabl
AUTHOR
Werner Schulte, May 23 2019
STATUS
approved

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Last modified April 14 09:38 EDT 2024. Contains 371657 sequences. (Running on oeis4.)