A195204 Triangle of coefficients of a sequence of binomial type polynomials.

2, 2, 4, 6, 12, 8, 26, 60, 48, 16, 150, 380, 360, 160, 32, 1082, 2940, 3120, 1680, 480, 64, 9366, 26908, 31080, 19040, 6720, 1344, 128, 94586, 284508, 351344, 236880, 96320, 24192, 3584, 256

Offset: 1

Author

Peter Bala, Sep 13 2011

Keywords

• nonn
• easy
• tabl

Comments

Define a polynomial sequence P_n(x) by means of the recursion

P_(n+1)(x) = x*(P_n(x)+ P_n(x+1)), with P_0(x) = 1.

The first few polynomials are

P_1(x) = 2*x, P_2(x) = 2*x*(2*x + 1),

P_3(x) = 2*x*(4*x^2 + 6*x + 3), P_4(x) = 2*x*(8*x^3+24*x^2+30*x+13).

The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. The P_n(x) are a polynomial sequence of binomial type. In particular, if we denote P_n(x) by x^[n] then we have the analog of the binomial expansion

(x+y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].

There are further analogies between the x^[n] and the monomials x^n.

1) Dobinski-type formula

exp(-x)*Sum_{k >= 0} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,2*x),

where the Bell (or exponential) polynomials are defined as

Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.

Equivalently, the connection constants associated with the polynomial sequences {x^[n]} and {x^n} are (up to signs) the same as the connection constants associated with the polynomial sequences {Bell(n,2*x)} and {Bell(n,x)}. For example, the list of coefficients of x^[4] is [26,60,48,16] and a calculation gives

Bell(4,2*x) = -26*Bell(1,x) + 60*Bell(2,x) - 48*Bell(3,x) + 16*Bell(4,x).

2) Analog of Bernoulli's summation formula

Bernoulli's formula for the sum of the p-th powers of the first n positive integers is

Sum_{k = 1..n} k^p = (1/(p+1))*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^(p+1-k), where B_k = [1,-1/2,1/6,0,-1/30,...] is the sequence of Bernoulli numbers.

This generalizes to

2*Sum_{k = 1..n} k^[p] = 1/(p+1)*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^[p+1-k].

The polynomials P_n(x) belong to a family of polynomial sequences P_n(x,t) of binomial type, dependent on a parameter t, and defined recursively by P_(n+1)(x,t)= x*(P_n(x,t)+ t*P_n(x+1,t)), with P_0(x,t) = 1. When t = 0 we have P_n(x,0) = x^n, the monomial polynomials. The present table is the case t = 1. The case t = -2 is (up to signs) A079641. See also A195205 (case t = 2).

Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011

T(n,k) is the number of binary relations R on [n] with index = 1 containing exactly k strongly connected components (SCC's) and satisfying the condition that if (x,y) is in R then x and y are in the same SCC. - Geoffrey Critzer, Jan 17 2024

Examples

			Triangle begins
n\k|....1......2......3......4......5......6......7
===================================================
..1|....2
..2|....2......4
..3|....6.....12......8
..4|...26.....60.....48.....16
..5|..150....380....360....160.....32
..6|.1082...2940...3120...1680....480.....64
..7|.9366..26908..31080..19040...6720...1344....128
...
Relation with rising factorials for row 4:
x^[4] = 16*x^4+48*x^3+60*x^2+26*x = 2^4*x*(x+1)*(x+2)*(x+3)-6*2^3*x*(x+1)*(x+2)+7*2^2*x*(x+1)-2*x, where [1,7,6,1] is the fourth row of the triangle of Stirling numbers of the second kind A008277.
Generalized Dobinski formula for row 4:
exp(-x)*Sum_{k >= 1} (-k)^[4]*x^k/k! = exp(-x)*Sum_{k >= 1} (16*k^4-48*k^3+60*k^2-26*k)*x^k/k! = 16*x^4+48*x^3+28*x^2+2*x = Bell(4,2*x).
Example of generalized Bernoulli summation formula:
2*(1^[2]+2^[2]+...+n^[2]) = 1/3*(B_0*n^[3]-3*B_1*n^[2]+3*B_2*n^[1]) =
n*(n+1)*(4*n+5)/3, where B_0 = 1, B_1 = -1/2, B_2 = 1/6 are Bernoulli numbers.
From _Philippe Deléham_, Dec 22 2011: (Start)
Triangle (0, 1, 2, 2, 4, 3, 6, ...) DELTA (2, 0, 2, 0, 2, ...) begins:
  1;
  0,    2;
  0,    2,     4;
  0,    6,    12,     8;
  0,   26,    60,    48,    16;
  0,  150,   380,   360,   160,   32;
  0, 1082,  2940,  3120,  1680,  480,   64;
  0, 9366, 26908, 31080, 19040, 6720, 1344, 128;
  ... (End)
		

Crossrefs

Cf. A000629 (row sums), A000670 (one half row sums), A014307 (row polys. at x = 1/2), A079641, A195205, A209849.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^(n+1)*polylog(-n, 2), 10); # Peter Luschny, Jan 29 2016

Mathematica

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(-1)^(#+1) PolyLog[-#, 2]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Formula

E.g.f.: F(x,z) := (exp(z)/(2-exp(z)))^x = Sum_{n>=0} P_n(x)*z^n/n!

= 1 + 2*x*z + (2*x+4*x^2)*z^2/2! + (6*x+12*x^2+8*x^3)*z^3/3! + ....

The generating function F(x,z) satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + x*F(x+1,z) and hence the row polynomials P_n(x) satisfy the recurrence relation

P_(n+1)(x)= x*(P_n(x) + P_n(x+1)), with P_0(x) = 1.

In what follows we change notation and write x^[n] for P_n(x).

Relation with the factorial polynomials:

For n >= 1,

x^[n] = Sum_{k = 1..n} (-1)^(n-k)*Stirling2(n,k)*2^k*x^(k),

and its inverse formula

2^n*x^(n) = Sum_{k = 1..n} |Stirling1(n,k)|*x^[k],

where x^(n) denotes the rising factorial x*(x+1)*...*(x+n-1).

Relation with the Bell polynomials:

The alternating n-th row entries (-1)^(n+k)*T(n,k) are the connection coefficients expressing the polynomial Bell(n,2*x) as a linear combination of Bell(k,x), 1 <= k <= n.

The delta operator:

The sequence of row polynomials is of binomial type. If D denotes the derivative operator d/dx then the delta operator D* for this sequence of binomial type polynomials is given by

D* = D/2 - log(cosh(D/2)) = log(2*exp(D)/(exp(D)+1))

= (D/2) - (D/2)^2/2! + 2*(D/2)^4/4! - 16*(D/2)^6/6! + 272*(D/2)^8/8! - ...,

where [1,2,16,272,...] is the sequence of tangent numbers A000182.

D* is the lowering operator for the row polynomials

(D*)x^[n] = n*x^[n-1].

Associated Bernoulli polynomials:

Generalized Bernoulli polynomial GB(n,x) associated with the polynomials x^[n] may be defined by

GB(n,x) := ((D*)/(exp(D)-1))x^[n].

They satisfy the difference equation

GB(n,x+1) - GB(n,x) = n*x^[n-1]

and have the expansion

GB(n,x) = -(1/2)*n*x^[n-1] + (1/2)*Sum_{k = 0..n} binomial(n,k) * B_k * x^[n-k], where B_k denotes the ordinary Bernoulli numbers.

The first few polynomials are

GB(0,x) = 1/2, GB(1,x) = x-3/4, GB(2,x) = 2*x^2-2*x+1/12,

GB(3,x) = 4*x^3-3*x^2-x, GB(4,x) = 8*x^4-4*x^2-4*x-1/60.

It can be shown that

1/(n+1)*(d/dx)(GB(n+1,x)) = Sum_{i = 0..n} 1/(i+1) * Sum_{k = 0..i} (-1)^k *binomial(i,k)*(x+k)^[n].

This generalizes a well-known formula for Bernoulli polynomials.

Relations with other sequences:

Row sums: A000629(n) = 2*A000670(n). Column 1: 2*A000670(n-1). Row polynomials evaluated at x = 1/2: {P_n(1/2)}n>=0 = [1,1,2,7,35,226,...] = A014307.

T(n,k) = A184962(n,k)*2^k. - Philippe Deléham, Feb 17 2013

Also the Bell transform of A076726. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x*z/(1 - z/(1 - 2*(x + 1)*z/(1 - 2*z/(1 - 2*(x + 2)*z/(1 - 3*z/(1 - 2*(x + 3)*z/(1 - 4*z/(1 - ... ))))))))). - Peter Bala, Dec 12 2024

Extensions

a(1) added by Philippe Deléham, Dec 22 2011