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A195205 Triangle of coefficients of a sequence of binomial type polynomials.

%I A195205 #35 Dec 16 2024 08:52:38
%S A195205 3,6,9,30,54,27,222,468,324,81,2190,5130,4320,1620,243,27006,68400,
%T A195205 65610,30780,7290,729,399630,1076166,1135890,618030,187110,30618,2187,
%U A195205 6899262,19532268,22212792,13471920,4796820,1020600,122472,6561
%N A195205 Triangle of coefficients of a sequence of binomial type polynomials.
%C A195205 Define a polynomial sequence P_n(x) by means of the recursion
%C A195205 P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1.
%C A195205 The first few values are
%C A195205 P_1(x) = 3*x, P_2(x) = 3*x*(3*x + 2),
%C A195205 P_3(x) = 3*x*(9*x^2 + 18*x + 10),
%C A195205 P_4(x) = 3*x*(27*x^3 + 108*x^2 + 156*x + 74).
%C A195205 The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. Compare with A195204.
%C A195205 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 A195205 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binomial_type">Binomial type</a>
%F A195205 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 A195205 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 A195205 Relation with rising factorials
%F A195205 x^[n] = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*3^k*x*(x+1)*...*(x+k-1),
%F A195205 and its inverse formula
%F A195205 3^n*x*(x+1)*...*(x+n-1) = Sum_{k=1..n} |Stirling1(n,k)|*x^[k].
%F A195205 The delta operator D*:
%F A195205 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 A195205 Generalized Dobinski formula:
%F A195205 exp(-x)*Sum_{k >= 1} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,3*x),
%F A195205 where the Bell (or exponential) polynomials are defined as
%F A195205 Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
%F A195205 Relation with the Bell polynomials:
%F A195205 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 A195205 Bell(4,3*x) = -222*Bell(1,x) + 468*Bell(2,x) - 324*Bell(3,x) + 81*Bell(4,x).
%F A195205 Generalized Bernoulli summation formula:
%F A195205 We have the following generalization of Bernoulli's formula for the sum of the powers of integers:
%F A195205 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 A195205 Relation with other sequences:
%F A195205 Row sums = 3*A050351(n) for n >= 1. Column 1 = 3*A004123.
%F A195205 T(n,k) = A185285(n,k)*3^k. - _Philippe Deléham_, Feb 17 2013
%F A195205 Also the Bell transform of 3*A004123. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%F A195205 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 A195205 Triangle begins
%e A195205 n\k|.....1.......2......3......4......5......6
%e A195205 ==============================================
%e A195205 ..1|.....3
%e A195205 ..2|.....6.......9
%e A195205 ..3|....30......54.....27
%e A195205 ..4|...222.....468....324.....81
%e A195205 ..5|..2190....5130...4320...1620....243
%e A195205 ..6|.27006...68400..65610..30780...7290....729
%e A195205 ...
%e A195205 Triangle (0, 2, 3, 4, 6, 6, 9, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, ...) begins:
%e A195205 1;
%e A195205 0,     3;
%e A195205 0,     6,     9;
%e A195205 0,    30,    54,    27;
%e A195205 0,   222,   468,   324,    81;
%e A195205 0,  2190,  5130,  4320,  1620,   243;
%e A195205 0, 27006, 68400, 65610, 30780,  7290,   729;
%e A195205 ... - _Philippe Deléham_, Dec 22 2011
%p A195205 # The function BellMatrix is defined in A264428.
%p A195205 # Adds (1,0,0,0, ..) as column 0.
%p A195205 BellMatrix(n -> `if`(n=0,3,polylog(-n, 2/3)), 10); # _Peter Luschny_, Jan 29 2016
%t A195205 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A195205 rows = 10;
%t A195205 M = BellMatrix[If[# == 0, 3, PolyLog[-#, 2/3]]&, rows];
%t A195205 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%Y A195205 Cf. A004123 ((1/3)*column 1), A050351 ((1/3)*row sums), A179929 (delta operator coeffs.), A195204.
%K A195205 nonn,easy,tabl
%O A195205 1,1
%A A195205 _Peter Bala_, Sep 13 2011