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A195204
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Triangle of coefficients of a sequence of binomial type polynomials.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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..inf} (-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
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LINKS
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Table of n, a(n) for n=1..36.
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FORMULA
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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 - ln(cosh(D/2)) = ln(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
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EXAMPLE
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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..inf} (-k)^[4]*x^k/k! = exp(-x)*sum {k = 1..inf} (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.
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 - Philippe Deléham, Dec 22 2011
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CROSSREFS
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Cf. A000629 (row sums), A000670(one half row sums), A014307 (row polys. at x = 1/2), A079641, A195205, A209849.
Sequence in context: A000672 A115868 A103299 * A154779 A010101 A129860
Adjacent sequences: A195201 A195202 A195203 * A195205 A195206 A195207
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Peter Bala, Sep 13 2011
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EXTENSIONS
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a(1) added by Philippe Deléham, Dec 22 2011
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STATUS
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approved
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