A195205 - OEIS

%I #35 Dec 16 2024 08:52:38

%S 3,6,9,30,54,27,222,468,324,81,2190,5130,4320,1620,243,27006,68400,

%T 65610,30780,7290,729,399630,1076166,1135890,618030,187110,30618,2187,

%U 6899262,19532268,22212792,13471920,4796820,1020600,122472,6561

%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) + 2*P_n(x+1)), with P_0(x) = 1.

%C The first few values are

%C P_1(x) = 3*x, P_2(x) = 3*x*(3*x + 2),

%C P_3(x) = 3*x*(9*x^2 + 18*x + 10),

%C P_4(x) = 3*x*(27*x^3 + 108*x^2 + 156*x + 74).

%C The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. Compare with A195204.

%C Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 22 2011

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binomial_type">Binomial type</a>

%F E.g.f.: F(x,z) := (exp(z)/(3 - 2*exp(z)))^x = 1 + 3*x*z + (6*x + 9*x^2)*z^2/2! + (30*x + 54*x^2 + 27*x^3)*z^3/3! + ....

%F The generating function F(x,z) = Sum_{n>=0} P_n(x)*z^n/n! satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + 2*x*F(x+1,z). Hence the row generating polynomials P_n(x) satisfy the recurrence P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1. The form of the e.g.f. shows that the polynomials P_n(x) are a sequence of binomial type. In what follows we denote P_n(x) by x^[n].

%F Relation with rising factorials

%F x^[n] = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*3^k*x*(x+1)*...*(x+k-1),

%F and its inverse formula

%F 3^n*x*(x+1)*...*(x+n-1) = Sum_{k=1..n} |Stirling1(n,k)|*x^[k].

%F The delta operator D*:

%F The row polynomials form a polynomial sequence of binomial type. If D denotes the derivative operator 1/3*d/dx then the associated delta operator D* is given by D* = D - 2*D^2/2! + 2*D^3/3! + 6*D^4/4! - 30*D^5/5! - ..., where the sequence of coefficients [1, -2, 2, 6, -30, -42, 882, ...] equals (-1)^n*A179929(n). D* is the lowering operator for the row polynomials, that is, (D*)x^[n] = n*x^[n-1].

%F Generalized Dobinski formula:

%F exp(-x)*Sum_{k >= 1} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,3*x),

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

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

%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,3*x) as a linear combination of Bell(k,x), 1 <= k <= n. For example for row 4:

%F Bell(4,3*x) = -222*Bell(1,x) + 468*Bell(2,x) - 324*Bell(3,x) + 81*Bell(4,x).

%F Generalized Bernoulli summation formula:

%F We have the following generalization of Bernoulli's formula for the sum of the powers of integers:

%F 3*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, ...] denotes the sequence of Bernoulli numbers.

%F Relation with other sequences:

%F Row sums = 3*A050351(n) for n >= 1. Column 1 = 3*A004123.

%F T(n,k) = A185285(n,k)*3^k. - _Philippe Deléham_, Feb 17 2013

%F Also the Bell transform of 3*A004123. 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 - 3*x*z/(1 - 2*z/(1 - 3*(x + 1)*z/(1 - 4*z/(1 - 3*(x + 2)*z/(1 - 6*z/(1 - 3*(x + 3)*z/(1 - 8*z/(1 - ... ))))))))). - _Peter Bala_, Dec 12 2024

%e Triangle begins

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

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

%e ..1|.....3

%e ..2|.....6.......9

%e ..3|....30......54.....27

%e ..4|...222.....468....324.....81

%e ..5|..2190....5130...4320...1620....243

%e ..6|.27006...68400..65610..30780...7290....729

%e ...

%e Triangle (0, 2, 3, 4, 6, 6, 9, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, ...) begins:

%e 1;

%e 0,     3;

%e 0,     6,     9;

%e 0,    30,    54,    27;

%e 0,   222,   468,   324,    81;

%e 0,  2190,  5130,  4320,  1620,   243;

%e 0, 27006, 68400, 65610, 30780,  7290,   729;

%e ... - _Philippe Deléham_, Dec 22 2011

%p # The function BellMatrix is defined in A264428.

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

%p BellMatrix(n -> `if`(n=0,3,polylog(-n, 2/3)), 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 = 10;

%t M = BellMatrix[If[# == 0, 3, PolyLog[-#, 2/3]]&, rows];

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

%Y Cf. A004123 ((1/3)*column 1), A050351 ((1/3)*row sums), A179929 (delta operator coeffs.), A195204.

%K nonn,easy,tabl

%O 1,1

%A _Peter Bala_, Sep 13 2011