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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-François Alcover, Feb 29 2016 *) PROG (PARI) /* q-Binomial coefficients: */ {C_q(n, k) = if(n

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Last modified August 6 18:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)