A023536 -id:A023536 - cp's OEIS frontend

A125811 Number of coefficients in the n-th q-Bell number as a polynomial in q.

1, 1, 1, 2, 3, 5, 8, 11, 15, 20, 26, 32, 39, 47, 56, 66, 76, 87, 99, 112, 126, 141, 156, 172, 189, 207, 226, 246, 267, 288, 310, 333, 357, 382, 408, 435, 463, 491, 520, 550, 581, 613, 646, 680, 715, 751, 787, 824, 862, 901, 941, 982, 1024, 1067, 1111, 1156, 1201

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			This sequence gives the number of terms in rows of A125810.
Row g.f.s B_q(n) of A125810 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.

Arvind Ayyer and Naren Sundaravaradan, An area-bounce exchanging bijection on a large subset of Dyck paths, arXiv:2401.14668 [math.CO], 2024. See p. 20.

Cf. A125810, A125812, A125813, A125814, A125815.

Cq:= proc(n,k) local j; if n nops(Bq(n)): seq(a(n), n=0..60); # Alois P. Heinz, Aug 04 2009

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; a[n_] := CoefficientList[QB[n, q], q] // Length; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Feb 29 2016 *)

/* q-Binomial coefficients: */
C_q(n,k)=if(n

from math import comb, isqrt
def A125811(n): return 1+comb(n,2)-sum(isqrt((k<<3)+1)-1>>1 for k in range(1,n)) # Chai Wah Wu, Feb 27 2025

a(n) = A023536(n-2) + 1.

a(n) = n*(n+1)/2 - 4 - Sum_{k=2..n-2} floor(1/2 + sqrt(2*k+4)) for n>2. [Due to a formula by Jan Hagberg in A023536]

More terms from Alois P. Heinz, Aug 04 2009

A279681 Irregular triangle read by rows: possible numbers of diagonals of convex polyhedra having n vertices.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Offset: 4

Vladimir Letsko, Dec 16 2016

Let n>4 denote the number of vertices. The set of possible numbers of diagonals is the union of sets {(k-1)(n-k-4), ..., (k-1)(n-(k+6)/2)}, where 1 <= k <= floor((sqrt(8n-15)-5)/2), and the set {(k-1)(n-k-4), ..., (n-3)(n-4)/2}, where k = floor((sqrt(8n-15)-3)/2). Note that cardinalities of all sets of this union excluding the last one are consecutive triangular numbers.

			Triangle begins:
4  | 0;
5  | 0, 1;
6  | 0, 1, 2, 3;
7  | 0, 1, 2, 3, 4, 5, 6;
8  | 0, 2, 3, 4, 5, 6, 7,  8,  9, 10;
9  | 0, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15;
10 | 0, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21;

Vladimir Letsko, Table of rows of a(n)

Row lengths are in A023536.

Cf. A000217, A279015, A279019, A279022, A279620, A279647, A279679.

dm:=(n, k)->simplify((n-1)*n/2-(2*n-k-1)-(n-k)*(n-k-3)/2-2*(k-1)-(k+2)*(k-1)/2);
dM:=(n, k)->simplify((n-1)*n/2-2*n-k+3-(n-k)*(n-k-3)/2);
Dv:=proc(n) local k, DD; DD:={0}:for k from 2 to n/2-1 do
DD:=DD union {seq(i, i=dm(n, k)..dM(n, k))} od:
DD:=DD union {seq(i, i=dm(n, k-1)..(n-3)*(n-4)/2)}:
DD end;

cp's OEIS Frontend

A125811 Number of coefficients in the n-th q-Bell number as a polynomial in q.

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