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

%I A195204 #45 Dec 16 2024 08:52:52
%S A195204 2,2,4,6,12,8,26,60,48,16,150,380,360,160,32,1082,2940,3120,1680,480,
%T A195204 64,9366,26908,31080,19040,6720,1344,128,94586,284508,351344,236880,
%U A195204 96320,24192,3584,256
%N A195204 Triangle of coefficients of a sequence of binomial type polynomials.
%C A195204 Define a polynomial sequence P_n(x) by means of the recursion
%C A195204 P_(n+1)(x) = x*(P_n(x)+ P_n(x+1)), with P_0(x) = 1.
%C A195204 The first few polynomials are
%C A195204 P_1(x) = 2*x, P_2(x) = 2*x*(2*x + 1),
%C A195204 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 A195204 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 A195204 (x+y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
%C A195204 There are further analogies between the x^[n] and the monomials x^n.
%C A195204 1) Dobinski-type formula
%C A195204 exp(-x)*Sum_{k >= 0} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,2*x),
%C A195204 where the Bell (or exponential) polynomials are defined as
%C A195204 Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
%C A195204 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 A195204 Bell(4,2*x) = -26*Bell(1,x) + 60*Bell(2,x) - 48*Bell(3,x) + 16*Bell(4,x).
%C A195204 2) Analog of Bernoulli's summation formula
%C A195204 Bernoulli's formula for the sum of the p-th powers of the first n positive integers is
%C A195204 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 A195204 This generalizes to
%C A195204 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 A195204 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 A195204 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 A195204 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 A195204 E.g.f.: F(x,z) := (exp(z)/(2-exp(z)))^x = Sum_{n>=0} P_n(x)*z^n/n!
%F A195204 = 1 + 2*x*z + (2*x+4*x^2)*z^2/2! + (6*x+12*x^2+8*x^3)*z^3/3! + ....
%F A195204 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 A195204 P_(n+1)(x)= x*(P_n(x) + P_n(x+1)), with P_0(x) = 1.
%F A195204 In what follows we change notation and write x^[n] for P_n(x).
%F A195204 Relation with the factorial polynomials:
%F A195204 For n >= 1,
%F A195204 x^[n] = Sum_{k = 1..n} (-1)^(n-k)*Stirling2(n,k)*2^k*x^(k),
%F A195204 and its inverse formula
%F A195204 2^n*x^(n) = Sum_{k = 1..n} |Stirling1(n,k)|*x^[k],
%F A195204 where x^(n) denotes the rising factorial x*(x+1)*...*(x+n-1).
%F A195204 Relation with the Bell polynomials:
%F A195204 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 A195204 The delta operator:
%F A195204 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 A195204 D* = D/2 - log(cosh(D/2)) = log(2*exp(D)/(exp(D)+1))
%F A195204 = (D/2) - (D/2)^2/2! + 2*(D/2)^4/4! - 16*(D/2)^6/6! + 272*(D/2)^8/8! - ...,
%F A195204 where [1,2,16,272,...] is the sequence of tangent numbers A000182.
%F A195204 D* is the lowering operator for the row polynomials
%F A195204 (D*)x^[n] = n*x^[n-1].
%F A195204 Associated Bernoulli polynomials:
%F A195204 Generalized Bernoulli polynomial GB(n,x) associated with the polynomials x^[n] may be defined by
%F A195204 GB(n,x) := ((D*)/(exp(D)-1))x^[n].
%F A195204 They satisfy the difference equation
%F A195204 GB(n,x+1) - GB(n,x) = n*x^[n-1]
%F A195204 and have the expansion
%F A195204 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 A195204 The first few polynomials are
%F A195204 GB(0,x) = 1/2, GB(1,x) = x-3/4, GB(2,x) = 2*x^2-2*x+1/12,
%F A195204 GB(3,x) = 4*x^3-3*x^2-x, GB(4,x) = 8*x^4-4*x^2-4*x-1/60.
%F A195204 It can be shown that
%F A195204 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 A195204 This generalizes a well-known formula for Bernoulli polynomials.
%F A195204 Relations with other sequences:
%F A195204 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 A195204 T(n,k) = A184962(n,k)*2^k. - _Philippe Deléham_, Feb 17 2013
%F A195204 Also the Bell transform of A076726. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%F A195204 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 A195204 Triangle begins
%e A195204 n\k|....1......2......3......4......5......6......7
%e A195204 ===================================================
%e A195204 ..1|....2
%e A195204 ..2|....2......4
%e A195204 ..3|....6.....12......8
%e A195204 ..4|...26.....60.....48.....16
%e A195204 ..5|..150....380....360....160.....32
%e A195204 ..6|.1082...2940...3120...1680....480.....64
%e A195204 ..7|.9366..26908..31080..19040...6720...1344....128
%e A195204 ...
%e A195204 Relation with rising factorials for row 4:
%e A195204 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 A195204 Generalized Dobinski formula for row 4:
%e A195204 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 A195204 Example of generalized Bernoulli summation formula:
%e A195204 2*(1^[2]+2^[2]+...+n^[2]) = 1/3*(B_0*n^[3]-3*B_1*n^[2]+3*B_2*n^[1]) =
%e A195204 n*(n+1)*(4*n+5)/3, where B_0 = 1, B_1 = -1/2, B_2 = 1/6 are Bernoulli numbers.
%e A195204 From _Philippe Deléham_, Dec 22 2011: (Start)
%e A195204 Triangle (0, 1, 2, 2, 4, 3, 6, ...) DELTA (2, 0, 2, 0, 2, ...) begins:
%e A195204   1;
%e A195204   0,    2;
%e A195204   0,    2,     4;
%e A195204   0,    6,    12,     8;
%e A195204   0,   26,    60,    48,    16;
%e A195204   0,  150,   380,   360,   160,   32;
%e A195204   0, 1082,  2940,  3120,  1680,  480,   64;
%e A195204   0, 9366, 26908, 31080, 19040, 6720, 1344, 128;
%e A195204   ... (End)
%p A195204 # The function BellMatrix is defined in A264428.
%p A195204 # Adds (1,0,0,0, ..) as column 0.
%p A195204 BellMatrix(n -> (-1)^(n+1)*polylog(-n, 2), 10); # _Peter Luschny_, Jan 29 2016
%t A195204 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A195204 rows = 12;
%t A195204 M = BellMatrix[(-1)^(#+1) PolyLog[-#, 2]&, rows];
%t A195204 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%Y A195204 Cf. A000629 (row sums), A000670 (one half row sums), A014307 (row polys. at x = 1/2), A079641, A195205, A209849.
%K A195204 nonn,easy,tabl
%O A195204 1,1
%A A195204 _Peter Bala_, Sep 13 2011
%E A195204 a(1) added by _Philippe Deléham_, Dec 22 2011