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A125810 Triangle of q-Bell number coefficients, read by rows that form polynomials in q, giving the eigensequence for the triangle of q-binomial coefficients. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row n evaluated at sample values of q are as follows:

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,k) is the number of set partitions of [n] having exactly k inversions. T(5,4)=3: 145|23, 145|2|3, 15|24|3; T(6,6) = 10: 1456|23, 156|234, 156|23|4, 1456|2|3, 146|25|3, 16|245|3, 156|2|34, 16|25|34, 156|2|3|4, 16|25|3|4. - Alois P. Heinz, Apr 03 2016

LINKS

Alois P. Heinz, Rows n = 0..50, flattened

FORMULA

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)].

Sum_{k>0} k * T(n,k) = A264082(n). - Alois P. Heinz, Apr 03 2016

EXAMPLE

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;

512, 1024, 2016, 3136, 4626, 6020, 7642, 8849, 9963, 10423, 10587, 10066, 9355, 8103, 6828, 5380, 4101, 2882, 1964, 1194, 708, 353, 167, 57, 18, 1; ...

MATHEMATICA

QB[n_, q_] := QB[n, q] = Sum[QB[j, q] QBinomial[n-1, j, q], {j, 0, n-1}] // FunctionExpand // Simplify; QB[0, q_]=1; QB[1, q_]=1; Table[ CoefficientList[QB[n, q], q], {n, 0, 9}] // Flatten (* Jean-Fran├žois Alcover, Feb 29 2016 *)

PROG

(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(" "))

CROSSREFS

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

Cf. A264082.

Sequence in context: A207605 A112931 A121685 * A226504 A152195 A133156

Adjacent sequences:  A125807 A125808 A125809 * A125811 A125812 A125813

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Dec 10 2006

STATUS

approved

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Last modified September 24 01:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)