0,3

Row n evaluated at sample values of q are: R_n(q=1) = A000110(n) (Bell numbers); R_n(q=-1) = A080107(n) (fixed points of permutation of SetPartitions); R_n(q=2) = A125812; R_n(q=3) = A125813; R_n(q=4) = A125814; R_n(q=5) = A125815.

T(n,0) = 2^(n-1) for n>0. G.f. of row n is a polynomial in q, B_q(n), that is generated by the recurrence: B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0, with B_q(0)=1. The q-binomial coefficient (also called Gaussian binomial coefficient) is given by: C_q(n,k) = [Product_{i=n-k+1..n} (1-q^i)]/[Product_{j=1..k} (1-q^j)].

Row g.f.s B_q(n) are polynomials in q generated by:

B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0 with B_q(0)=1

where the triangle of q-binomial coefficients C_q(n,k) begins:

1;

1, 1;

1, 1 + q, 1;

1, 1 + q + q^2, 1 + q + q^2, 1;

1, 1 + q + q^2 + q^3, 1 + q + 2*q^2 + q^3 + q^4, 1 + q + q^2 + q^3, 1;

The initial q-Bell coefficients in B_q(n) are:

B_q(0) = 1; B_q(1) = 1; B_q(2) = 2;

B_q(3) = 4 + q;

B_q(4) = 8 + 4*q + 3*q^2;

B_q(5) = 16 + 12*q + 13*q^2 + 8*q^3 + 3*q^4;

B_q(6) = 32 + 32*q + 42*q^2 + 38*q^3 + 33*q^4 + 15*q^5 + 10*q^6 + q^7.

Number of terms in row n is given by A125811, which starts:

1,1,1,2,3,5,8,11,15,20,26,32,39,47,56,66,76,87,99,112,126,141,156,...

Triangle begins:

1;

1;

2;

4, 1;

8, 4, 3;

16, 12, 13, 8, 3;

32, 32, 42, 38, 33, 15, 10, 1;

64, 80, 120, 133, 145, 121, 98, 60, 37, 15, 4;

128, 192, 320, 408, 507, 526, 544, 457, 391, 281, 195, 104, 61, 20, 6;

256, 448, 816, 1160, 1585, 1875, 2189, 2259, 2256, 2066, 1819, 1450, 1133, 777, 506, 300, 158, 65, 25, 4; ...

(PARI) /* q-Binomial coefficients: */ {C_q(n, k)=if(n<k|k<0, 0, if(n==0|k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j)))} /* q-Bell numbers = eigensequence of q-binomial triangle: */ {B_q(n)=if(n==0, 1, sum(k=0, n-1, B_q(k)*C_q(n-1, k)))} /* Coefficients in row n: */ {T(n, k)=polcoeff(B_q(n), k, q)} /* Print triangle rows: */ for(n=0, 10, for(k=0, #Vec(B_q(n))-1, print1(T(n, k), ", ")); print(" "))

Cf. A000110, A080107, A125811, A125812, A125813, A125814, A125815.

Sequence in context: A065278 A112931 A121685 this_sequence A133156 A127529 A091977

Adjacent sequences: A125807 A125808 A125809 this_sequence A125811 A125812 A125813

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2006