This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A185419 #32 Mar 23 2020 17:31:49
%S A185419 1,3,1,10,9,1,42,67,18,1,248,510,235,30,1,1992,4378,2835,605,45,1,
%T A185419 19600,44268,34888,10605,1295,63,1,222288,524748,461748,178913,31080,
%U A185419 2450,84,1,2851712,7103088,6728428,3069612,690753,77112,4242,108,1
%N A185419 Table of coefficients of a polynomial sequence of binomial type related to the enumeration of minimax trees A080795.
%C A185419 DEFINITION
%C A185419 Define a sequence of polynomials M(n,x) by means of the recurrence relation
%C A185419 (1)... M(n+1,x) = x*{2*M(n,x+1)-M(n,x-1)},
%C A185419 with starting value M(0,x) = 1. We call these the minimax polynomials.
%C A185419 The first few polynomials are
%C A185419 M(1,x) = x
%C A185419 M(2,x) = x*(x + 3)
%C A185419 M(3,x) = x*(x^2 + 9*x + 10)
%C A185419 M(4,x) = x*(x^3 + 18*x^2 + 67*x + 42)
%C A185419 M(5,x) = x*(x^4 + 30*x^3 + 235*x^2 + 510*x + 248).
%C A185419 This triangle lists the coefficients of these polynomials (apart from M(0,x)) in ascending powers of x.
%C A185419 RELATION TO MINIMAX TREES
%C A185419 The value M(n,1) equals the number of minimax trees on n nodes - A080795(n). This result can be used to recursively calculate the entries of A080795 - see A185420.
%C A185419 In addition, the minimax polynomials M(n,x) occur in the formula for the number T(n,k) of forests of k minimax trees on n nodes. ... T(n,k) = 1/k!*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*M(n,j).
%C A185419 ANALOGIES WITH THE MONOMIALS
%C A185419 {M(n,x)}n>=0 is a polynomial sequence of binomial type and so is analogous to the sequence of monomials x^n. Denoting M(n,x) by x^[n] to emphasize this analogy, we have, for example, the following analog of Bernoulli's formula for the sum of integer powers:
%C A185419 (2)... 1^[p]+...+(n-1)^[p] = -2*n^[p]+ 1/(p+1)*Sum_{k = 0..floor(p/2)} 8^k*binomial(p+1,2k)*B_(2k)*n^[p+1-2k], where {B_k}k>=0 = [1, -1/2, 1/6, 0, -1/30, ...] is the sequence of Bernoulli numbers.
%C A185419 For other polynomial sequences defined by recurrences similar to (1), and related to the zigzag numbers A000111 and the Springer numbers A001586, see A147309 and A185417, respectively. See also A185415.
%C A185419 The Bell transform of A143523(n). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%F A185419 GENERATING FUNCTION
%F A185419 Let a = 3-2*sqrt(2). Let f(t) = (1/2)*sqrt(2)*((1+a*exp(2*sqrt(2)*t))/ (1-a*exp(2*sqrt(2)*t))) = 1 + t + 4*t^2/2! + 20*t^3/3! + ... be the e.g.f. for A080795. Then the e.g.f. for the current table, including a constant 1, is
%F A185419 (1)... F(x,t) = f(t)^x = Sum_{n>=0} M(n,x)*t^n/n! = 1 + x*t + (3*x+x^2)*t^2/2! + (10*x+9*x^2+x^3)*t^3/3! + ....
%F A185419 ROW POLYNOMIALS
%F A185419 One easily checks that d/dt(F(x,t)) = x*(2*F(x+1,t)-F(x-1,t)) and hence the row generating polynomials M(n,x) satisfy the recurrence relation
%F A185419 (2)... M(n+1,x) = x*{2*M(n,x+1)-M(n,x-1)}.
%F A185419 The form of the e.g.f shows that the row polynomials are a polynomial sequence of binomial type. The associated delta operator D* is given by
%F A185419 (3)... D* = sqrt(2)/4*log((3+2*sqrt(2))*(sqrt(2)*exp(D)-1)/(sqrt(2)*exp(D)+1)),
%F A185419 where D is the derivative operator d/dx. This expands to
%F A185419 (4)... D* = D - 3*D^2/2! + 17*D^3/3! - 147*D^4/4! + ....
%F A185419 The sequence of coefficients [1,3,17,147,...] is A080253.
%F A185419 The delta operator D* acts as a lowering operator for the minimax polynomials
%F A185419 (5)...(D*) M(n,x) = n*M(n-1,x).
%F A185419 In what follows it will be convenient to denote M(n,x) by x^[n].
%F A185419 ANALOG OF THE LITTLE FERMAT THEOREM
%F A185419 For integer x and odd prime p
%F A185419 (6)... x^[p] = (-1)^((p^2-1)/8)*x (mod p).
%F A185419 More generally, for k = 1,2,...
%F A185419 (7)... x^[p+k-1] = (-1)^((p^2-1)/8)*x^[k] (mod p).
%F A185419 GENERALIZED BERNOULLI POLYNOMIALS ASSOCIATED WITH THE MINIMAX POLYNOMIALS
%F A185419 The generalized Bernoulli polynomial MB(k,x) associated with the minimax polynomial x^[k] (= M(k,x)) may be defined as the result of applying the differential operator D*/(exp(D)-1) to the polynomial x^[k]:
%F A185419 (8)... MB(k,x) := {D*/(exp(D)-1)} x^[k].
%F A185419 The first few generalized Bernoulli polynomials are
%F A185419 MB(0,x) = 1,
%F A185419 MB(1,x) = x - 2,
%F A185419 MB(2,x) = x^2 - x + 4/3,
%F A185419 MB(3,x) = x^3 + 3*x^2 - 4*x,
%F A185419 MB(4,x) = x^4 + 10*x^3 + 3*x^2 - 14*x - 32/15.
%F A185419 Since exp(D)-1 is the forward difference operator it follows from (5) and (8) that
%F A185419 (9)... MB(k,x+1) - MB(k,x) = k*x^[k-1].
%F A185419 Summing (9) from x = 1 to x = n-1 and telescoping we find a closed form expression for the finite sums
%F A185419 (10)... 1^[p]+2^[p]+...+(n-1)^[p] = 1/(p+1)*{MB(p+1,n)-MB(p+1,1)}.
%F A185419 The generalized Bernoulli polynomials can be expanded in terms of the minimax polynomials x^[k]. Use (3) to express exp(D)-1 in terms of D*.
%F A185419 Substitute the resulting expression in (8) and expand as a power series in D* to arrive at the expansion:
%F A185419 (11)... MB(k,x) = -2*k*x^[k-1] + Sum_{j=0..floor(k/2)} 2^(3*j) * binomial(k,2j)*B_(2j)*x^[k-2j], where {B_j}j>=0 = [1,-1/2,1/6,0,-1/30,...] denotes the Bernoulli number sequence.
%F A185419 RELATION WITH OTHER SEQUENCES
%F A185419 Column 1 [1, 3, 10, 42, 248, ...] = A143523 with an offset of 1.
%F A185419 Row sums [1, 1, 4, 20, 128, 1024, ...] = A080795.
%e A185419 Triangle begins
%e A185419 n\k|.....1......2......3......4......5......6......7
%e A185419 ====================================================
%e A185419 ..1|.....1
%e A185419 ..2|.....3......1
%e A185419 ..3|....10......9......1
%e A185419 ..4|....42.....67.....18......1
%e A185419 ..5|...248....510....235.....30......1
%e A185419 ..6|..1992...4378...2835....605.....45......1
%e A185419 ..7|.19600..44268..34888..10605...1295.....63......1
%e A185419 ..
%e A185419 Example of the generalized Bernoulli summation formula:
%e A185419 The second row of the triangle gives x^[2] = 3*x+x^2.
%e A185419 Then 1^[2]+2^[2]+...+(n-1)^[2] = (n^3+3*n^2-4*n)/3 = 1/3*(MB(3,n)-MB(3,0)).
%e A185419 From _R. J. Mathar_, Mar 15 2013: (Start)
%e A185419 The matrix inverse starts
%e A185419 1;
%e A185419 -3, 1;
%e A185419 17, -9, 1;
%e A185419 -147, 95, -18, 1;
%e A185419 1697, -1245, 305, -30, 1;
%e A185419 -24483, 19687, -5670, 745, -45, 1;
%e A185419 423857, -365757, 118237, -18690, 1540, -63, 1;
%e A185419 -8560947, 7819287, -2761122, 498197, -50190, 2842, -84, 1; (End)
%p A185419 #A185419
%p A185419 M := proc(n,x) option remember;
%p A185419 if n = 0 then
%p A185419 return 1
%p A185419 else return
%p A185419 x*(2*M(n-1,x+1)-M(n-1,x-1))
%p A185419 end if;
%p A185419 end proc:
%p A185419 with(PolynomialTools):
%p A185419 for n from 1 to 10 do
%p A185419 CoefficientList(M(n,x),x);
%p A185419 end do;
%t A185419 M[0, _] = 1; M[n_, x_] := M[n, x] = x (2 M[n-1, x+1] - M[n-1, x-1]);
%t A185419 Table[CoefficientList[M[n, x], x] // Rest, {n, 1, 10}] (* _Jean-François Alcover_, Jun 26 2019 *)
%o A185419 (Sage) # uses[bell_matrix from A264428]
%o A185419 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A185419 bell_matrix(lambda n: A143523(n), 10) # _Peter Luschny_, Jan 18 2016
%Y A185419 Cf. A080253 (coeffs. of delta operator), A080795 (row sums), A143523 (column 1), A147309, A185415, A185417, A185420.
%K A185419 nonn,easy,tabl
%O A185419 1,2
%A A185419 _Peter Bala_, Feb 07 2011