A195204 - OEIS

%I #45 Dec 16 2024 08:52:52

%S 2,2,4,6,12,8,26,60,48,16,150,380,360,160,32,1082,2940,3120,1680,480,

%T 64,9366,26908,31080,19040,6720,1344,128,94586,284508,351344,236880,

%U 96320,24192,3584,256

%N Triangle of coefficients of a sequence of binomial type polynomials.

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

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

%C The first few polynomials are

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

%C 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).

%C 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

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

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

%C 1) Dobinski-type formula

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

%C where the Bell (or exponential) polynomials are defined as

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

%C 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

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

%C 2) Analog of Bernoulli's summation formula

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

%C 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.

%C This generalizes to

%C 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].

%C 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).

%C 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

%C 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

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

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

%F 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

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

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

%F Relation with the factorial polynomials:

%F For n >= 1,

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

%F and its inverse formula

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

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

%F Relation with the Bell polynomials:

%F 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.

%F The delta operator:

%F 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

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

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

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

%F D* is the lowering operator for the row polynomials

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

%F Associated Bernoulli polynomials:

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

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

%F They satisfy the difference equation

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

%F and have the expansion

%F 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.

%F The first few polynomials are

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

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

%F It can be shown that

%F 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].

%F This generalizes a well-known formula for Bernoulli polynomials.

%F Relations with other sequences:

%F 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.

%F T(n,k) = A184962(n,k)*2^k. - _Philippe Deléham_, Feb 17 2013

%F Also the Bell transform of A076726. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016

%F 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

%e Triangle begins

%e n\k|....1......2......3......4......5......6......7

%e ===================================================

%e ..1|....2

%e ..2|....2......4

%e ..3|....6.....12......8

%e ..4|...26.....60.....48.....16

%e ..5|..150....380....360....160.....32

%e ..6|.1082...2940...3120...1680....480.....64

%e ..7|.9366..26908..31080..19040...6720...1344....128

%e ...

%e Relation with rising factorials for row 4:

%e 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.

%e Generalized Dobinski formula for row 4:

%e 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).

%e Example of generalized Bernoulli summation formula:

%e 2*(1^[2]+2^[2]+...+n^[2]) = 1/3*(B_0*n^[3]-3*B_1*n^[2]+3*B_2*n^[1]) =

%e n*(n+1)*(4*n+5)/3, where B_0 = 1, B_1 = -1/2, B_2 = 1/6 are Bernoulli numbers.

%e From _Philippe Deléham_, Dec 22 2011: (Start)

%e Triangle (0, 1, 2, 2, 4, 3, 6, ...) DELTA (2, 0, 2, 0, 2, ...) begins:

%e   1;

%e   0,    2;

%e   0,    2,     4;

%e   0,    6,    12,     8;

%e   0,   26,    60,    48,    16;

%e   0,  150,   380,   360,   160,   32;

%e   0, 1082,  2940,  3120,  1680,  480,   64;

%e   0, 9366, 26908, 31080, 19040, 6720, 1344, 128;

%e   ... (End)

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> (-1)^(n+1)*polylog(-n, 2), 10); # _Peter Luschny_, Jan 29 2016

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t rows = 12;

%t M = BellMatrix[(-1)^(#+1) PolyLog[-#, 2]&, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)

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

%K nonn,easy,tabl

%O 1,1

%A _Peter Bala_, Sep 13 2011

%E a(1) added by _Philippe Deléham_, Dec 22 2011