A270236 Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.
1, 3, 1, 9, 5, 1, 30, 21, 8, 1, 112, 88, 47, 12, 1, 463, 387, 253, 97, 17, 1, 2095, 1816, 1345, 675, 184, 23, 1, 10279, 9123, 7304, 4418, 1641, 324, 30, 1, 54267, 48971, 41193, 28396, 13276, 3645, 536, 38, 1, 306298, 279855, 243152, 183615, 102244, 36223, 7473, 842, 47, 1
Offset: 1
A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).
1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 1, 1, 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1, 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1, 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1, 1, 7, 21, 56, 105, 161, 231, 302, 356, 379, 392, 384, 358, 314, 262
Offset: 0
Examples
Number of terms in row n is: n*(n-1)/2 + 1. Row functions B(n,q) begin: B(0,q) = 1; B(1,q) = 1; B(2,q) = 1 + q; B(3,q) = 1 + 2*q + q^2 + q^3; B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6. Triangle begins: 1; 1; 1, 1; 1, 2, 1, 1; 1, 3, 3, 4, 2, 1, 1; 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1; 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1; 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1; ...
Links
- Alois P. Heinz, Rows n = 0..60, flattened
- Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), J. Integer Seqs., Vol. 7, 2004.
Programs
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Maple
b:= proc(n, m, t) option remember; `if`(n=0, x^t, add(b(n-1, max(m, j), t+j) , j=1..m+1)) end: T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)): seq(T(n), n=0..8); # Alois P. Heinz, Aug 02 2021
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Mathematica
B[0, ] = 1; B[n, q_] := B[n, q] = Sum[Binomial[n-1, k] B[k, q] q^k, {k, 0, n-1}] // Expand; Table[CoefficientList[B[n, q], q], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 08 2016 *)
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PARI
{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))} row(n)={Vec(B(n, 'q)+O('q^(n*(n-1)/2+1)))}
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PARI
/* Alternative formula for the n-th q-Bell number (row n): */ {B(n,q)=local(inf=100);round((0^n + sum(k=1, inf,((q^k-1)/(q-1))^n/prod(i=1,k,(q^i-1)/(q-1)))) / prod(k=1, inf,1 + (q-1)/q^k))}
Formula
G.f. for row n: B(n,q) = 1/E_q*{0^n + Sum_{k>=1} [(q^k-1)/(q-1)]^n / q-Factorial(k)}, where q-Factorial(k) = Product_{j=1..k} [(q^j-1)/(q-1)] and where E_q = Sum_{n>=0} 1/q-Factorial(n) = Product_{n>=1} (1+(q-1)/q^n).
Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - Alois P. Heinz, Aug 02 2021
Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025
Extensions
Keyword:tabl changed to tabf by R. J. Mathar, Oct 21 2010
A120057 Table T(n,k) = sum over all set partitions of n of number at index k.
1, 2, 3, 5, 8, 9, 15, 25, 29, 31, 52, 89, 106, 115, 120, 203, 354, 431, 474, 499, 514, 877, 1551, 1923, 2141, 2273, 2355, 2407, 4140, 7403, 9318, 10489, 11224, 11695, 12002, 12205, 21147, 38154, 48635, 55286, 59595, 62434, 64331, 65614, 66491, 115975, 210803, 271617, 311469, 338019, 355951, 368205, 376665, 382559, 386699
Offset: 1
Examples
The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}. Summing these componentwise gives the third row: 5,8,9. Table starts: 1; 2, 3; 5, 8, 9; 15, 25, 29, 31; 52, 89, 106, 115, 120; ...
Links
- Alois P. Heinz, Rows n = 1..141, flattened
Crossrefs
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [p[1], expand(p[2]*x+p[1]*j)])( b(n-1, max(m, j))), j=1..m+1)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0)[2]): seq(T(n), n=1..10); # Alois P. Heinz, Mar 24 2016
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Mathematica
b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*x + p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0][[2]]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 08 2016, after Alois P. Heinz *)
Comments
Examples
Links
Crossrefs
Programs
Maple
Mathematica
Formula